UC-NRLF 


THE  GALVANOMETER 


A  SERIES  OF  LECTURES 


BY 


EDWARD  L.   NICHOLS 


PROFESSOR    OF    PHYSICS     IN    CORNELL    UNIVERSITY 


As  printed  in  ELECTRIC  POWER. 


NEW  YORK 
McILROY    &    EMMET 

1894 


UNIVERSITY 


GALVANOMETER  AT  CORNELL  UNIVERSITY. 


4      MfV. 


THE  GALVANOMETER 


A  SERIES  OF  LECTURES 


BY 


EDWARD  L  NICHOLS 

»  V 

PROFESSOR   OF   PHYSICS  IN  CORNELL   UNIVERSITY 


OF  THE 


NEW  YORK 
McILROY    &    EMMET 

1894 


Entered  according  to  Act  of  Congress  in  the  year  1894  by 

McTLROY    &    EMMET, 
in  the  office  of  the  Librarian  of  Congress  at  Washington. 


XX 

f  OF  THE 

(UNIVERSITY 


'"PHIS  series  of  lectures  was  written,  in  the  first  place, 
for  the  benefit  of  a  class  of  students  of  electrical  engi- 
neering- in  Cornell  University.  In  it  I  have  endeavored 
to  bring  together,  in  compact  form,  for  the  benefit  of 
readers  of  limited  mathematical  attainments,  the  most 
important  features  of  the  theory  of  the  galvanometer, 
together  with  some  suggestions  concerning  the  methods 
of  using  that  instrument.  '  To  this  end,  free  use  has 
been  made  of  the  work  of  many  writers.  The  treatises 
of  Maxwell,  and  of  Mascart  and  Joubert,  in  particular, 
have  been  repeatedly  drawn  upon.  From  various  fel- 
low physicists,  also,  I  have  received  hints.  The  services 
of  one  of  these,  Professor  W.  S.  Franklin,  I  desire  es- 
pecially to  acknowledge,  since  several  important  features 
in  the  theoretical  treatment  of  my  subject  are  due  to 
suggestions  made  by  him.  Detailed  studies  of  the  per- 
formance of  sensitive  galvanometers  by  Professor 
Ernest  Merritt  and  Mr.  F.  J.  Rogers,  and  a  variety  of 
details  culled  from  the  records,  published  and  unpub- 
lished, of  other  workers  in  the  laboratories  of  the  de- 
partment of  physics  have  been  made  use  of. 

Nearly  all  that  has  been  written  about  galvanometers, 
aside  from  the  theory  and  details  of  construction,  applies 
to  the  types  of  instrument  which  were  called  into  ex- 
istence by  the  requirements  of  the  testing  of  cables  and 
telegraph  lines  and  of  submarine  signalling.  Concern- 
ing the  use  of  such  instruments  on  the  one  hand  and 
upon  the  subject  of  what  may  be  termed  voltmeter  and 
ammeter  work,  from  its  relation  to  the  practice  and  meth- 


ods  of  the  dynamo  laboratory,  abundant  material  is 
already  accessible  in  the  manuals  of  Kempe,  Ayrton, 
Kittler  and  many  other  writers.  To  this  part  of  the 
subject  I  have  paid  little  attention. 

The  extraordinary  demands  upon  the  sensitiveness  of 
the  galvanometer  made  by  the  researches  of  Langley, 
Aegstrom,  Julius,  Rubens,  Snow,  Paschen  and  others 
in  the  domain  of  radiation,  has,  however,  resulted  in  the 
development  of  a  new  class  of  instruments,  the  delicacy 
of  which  has  greatly  modified  the  art  of  using  the  gal- 
vanometer. Of  these  matters  I  have  endeavored  to  give 
some  account. 

EDWARD  L.  NICHOLS. 
January,  1894. 


TCTNIVERSITT 


LECTURE   I. 

Galvanometers  for  absolute  measurement. — Three  effects 
of  the  voltaic  current  may  be  made  use  of  for  measure- 
ment ;  the  thermal,  utilized  in  electro-calorimetry,  the 
chemical,  which  is  used  in  voltametry,  and  the  magnetic, 
upon  which  the  action  of  the  galvanometer  depends. 
The  essential  parts  of  this  instrument  are  a  magnet 
needle,  suspended  in  general  with  freedom  of  vibration 


FIG.    I. 


about  a  vertical  axis,  and  an  electric  circuit,  consisting 
of  one  or  more  coils  of  wire,  within  the  field  of  which 
the  needle  swings. 

For  purposes  of  absolute  measurement  the  coils  of 
the  galvanometer  must  be  of  known  dimensions,  they 
must  be  at  a  known  distance  from  the  needle,  and  the 
latter  must  be  situated  in  a  magnetic  field  of  known 
intensity.  The  simplest  form,  which  is  also  that  most 
frequently  met  with  in  practice,  consists  of  one  or  more 
circular  coils  mounted  vertically  in  the  magnetic  meri- 
dian. Where  a  single  coil  is  used  the  needle  is  in  its 
axis.  When  there  is  more  than  one  coil,  these  have  a 
common  axis  in  which  the  needle  hangs. 


The  law  of  action  of  the  galvanometer  may  be  de- 
rived from  the  following  familiar  and  well-established 
principles  : 

1.  Influence  of  a  current  upon  a  magnetic  particle. — Given 
a  conductor  //,  carrying  current  at  right  angles  to  the 
plane  of  the  paper  (Fig.  i)  in  the  direction  indicated  by 
the  lines  of  force.     A  particle  //  situated  in  the  field 
surrounding  L  will  tend  to  move  along  a  line  of  force. 

2.  Influence  of  a  circular  current. — If  i  is  a  circular  con- 
ductor and  p.  be  situated  in  the  axis  of  the  ring  (Fig.  2) 
at  a  distance  x  from  the  plane  of  the  latter,  and  at  a 
distance  d  from  the  conductor,  the  action  of  each  ele- 
ment (d  L)  of  the  ring  will  be  to  tend  to  drive  ft  along 


d  L 


FIG.    2. 


FIG.    3. 


A  B,  tangent  to  the  line  of  force  at  that  point  with  a 
force  f(d  L),  such  that — 

Const,  i  d  L  _  Const,  i  d  L 

J(d  L)  -  ff  ^  _|_  ^o  (1) 

For  the  entire  circle  the  force  fa)  is 


/(£)  =  Const,  ^-qj 


If  jj.  be  a  magnet-pole  of  strength  m  with  freedom  of 
motion  only  around  a  vertical  axis,  the  case  with  which 
we  have  to  do  in  considering  the  galvanometer,  the 
effective  component  of  fa)  along  the  axis  is 


ft    —  Const. 


fynriva 


=  Const. 


2  TT  r2  i  m 


.(3) 


When  r  and  x  are  taken  in  centimeters  the  constant  is 
unity  and  i  is  expressed  in  absolute  measure. 

The  actual  case  to  be  considered  is  that  of  a  needle, 
swinging  in  a  magnetic  field.  This  field  is  made  up  of 
two  active  components,  the  horizontal  force  (fe)  of  the 
earth's  magnetism,  or  of  an  artificial  field  which  takes 
its  place,  and  the  horizontal  component  of  the  force  due 
to  the  current  (/*•),  see  Fig.  3. 

When  the  couple  2  I  fe  sin  $,  due  to  the  action  of  the 
earth's  field  is  balanced  by  the  couple  2  I  //  cos  $,  the 
needle  is  in  equilibrium,  a  condition  which  is  expressed 
by  the  equation. 


/     /// 


X 


FIG.    4. 


from  which,  since  fe  =  Jim,  where  ^Tis  the  horizontal 
component  of  the  earth's  magnetism  and  ra  is  the 
strength  of  the  magnet  pole,  we  derive  the  equation  of 
the  tangent  galvanometer  of  a  single  turn. 


(5) 


If  the  galvanometer  consists  of  any  number  of  sepa- 
rate coils  the  radii  of  which  are  rlf  r2,  rs, 


etc.,  at  dis- 


tances  x^x2)  a*3,  x4)  etc.,  from  the  needle,  containing  re- 
spectively Wj,  ??2,  ft  3,  ??4,  etc.,  turns  of  wire,  each  coil  will 
have  its  independent  action  upon  the  needle,  and  the 
following  general  equation  of  the  galvanometer  may 
be  written.  This  equation  is  applicable  in  every  case 
in  which  the  coils  are  in  the  magnetic  meridian,  and 
possess  a  common  axis,  in  which  axis  the  needle,  the 
length  of  which  must  be  small  compared  with  the  radii 
of  the  coils,  is  suspended.  This  equation  is, 


2  7T 


"271 


(rf 


2  it  r 


+  etc. 


FIG.    5. 


The  denominator  of  the  right-hand   member  of  this 
equation  is  termed  the  constant  of  the  coils,  and  it  is  desig- 

TT 

nated  by  the  letter  G^;*  the  ratio  -^  is  the  constant  of  the 

galvanometer.     Equation  (6),  then,  may  be  written  either 
in  the  form 

*  The  reciprocal  of  this  expression  is  sometimes  taken  as  the  constant  of  the  coils, 
in  which  case  equation  (7)  is  written 

TT 

i  =  G  H  tan  $,  instead  of  i  =   -Q  tan  $• 


or  in  the  form 


i  =        tan  *.  (7) 

i  =  JTtan  #.  (8) 

As  a  matter  of  construction,  tangent  galvanometers 
usually  have  either  one  or  two  coils.  The  large  gal- 
vanometer of  Cornell  University  with  six  coils  is  really 
a  combination  of  several  instruments,  designed  for  heavy 
currents  and  one  for  currents  of  small  intensity. 

Of  galvanometers  with  a  single  coil,  the  common 
form  has  the  needle  in  the  plane  of  the  ring.  For  such 


FIG.    6. 

instruments   the   quantity  x  in    equation   (6)   is   zero. 
The  equation  then  takes  the  following  simpler  form  : 

i  =  ^-n  H  tan  &.  (9) 

The  objection  to  galvanometers  with  the  needle  in 
the  plane  of  the  ring  is  in  the  nature  of  the  field  of 
force  due  to  the  current  in  a  single  ring.  This  field  of 
a  circular  current  was  discussed  by  Lord  Kelvin  in 
1869  ;*  from  whose  results,  as  embodied  in  Plate  XVIII. 
of  the  second  volume  of  Maxwell's  Treatise  on  Electric- 
ity, Fig.  4  is  taken. 

*  W.  Thomson,  Transactions  of  the  Royal  Society  of  Edinburgh,  vol.  25,  p.  217. 


In  such  a  field,  by  the  use  of  a  long  needle  or  the  dis- 
placement of  the  needle  from  its  position  in  the  centre 
of  the  coil,  considerable  errors  are  introduced.  It  is  to 
reduce  such  errors  to  a  minimum  that  the  Helmholtz 
form  of  the  tangent  galvanometer  is  used.  In  this  in- 
strument two  coils  of  equal  area  are  placed  at  a  dis- 
tance apart  equal  to  their  radius,  and  the  needle  is. 


FIG.    7. 


situated  midway  between  them  in  their  common  axis. 
The  field  produced  by  current  in  coils  thus  located  is 
very  nearly  uniform  for  a  considerable  region  surround- 
ing the  centre  of  the  system,  and  displacements  of  the 
needle  have  of  but  slight  influence  upon  the  perform- 
ance of  the  galvanometer.  Figure  5,  also  from  Lord 


10 


Kelvin's  paper  just  cited,  affords  a  comparison  of  the 
field  of  two  parallel  circular  currents,  with  that  pro- 
duced by  a  single  coil. 

The  large  tangent  galvanometer  of  Cornell  Univer- 
sity, to  which  brief  reference  has  just  been  made,  affords 
an  interesting  example  of  the  application  of  the  Helm- 
holtz  construction.  A  diagram  of  this  instrument, 
showing  the  proportions  and  dimensions  of  the  various 
coils,  is  given  in  Fig.  6. 

It  consists,  essentially,  of  four  distinct  instruments  ar- 
ranged so  as  to  be  used  separately  or  in  combination. 
There  are  : 

a.  A  Helmholtz  galvanometer  (i,  4,  i,  4)  with  coils 
about  two  meters  in  diameter,  each  consisting  of  a  single 
turn  of  heavy  wire  of  about  2.00  cm.  diameter. 

b.  A  similar  galvanometer  symmetrically  placed  with 
reference  to  the  first  and  acting  upon  the  same  needle, 
with  coils  of  the  same  heavy  wire,  the  diameter  of  each 
coil  being  about  160  cm. 


FIG.  8. 

c.  A   Helmholtz  galvanometer,   for    small   currents, 
with  36  turns,  divided  into  two  sections  of  eighteen 
turns  apiece,  in   each  coil,  the  mean  diameter  of  the 
coils  about  152   cm.     The  centre  of  the  system  coin- 
cides with  that  of  a  and  b,  and  the  same  needle  serves. 

d.  A  modification  of  the  Kohlrausch  instrument  for 
the  determination  of  H.   This  consists  of  a  coil  100  cm. 
in  diameter  with  100  turns  of  wire.    It  is  suspended  in 
the  magnetic  meridian  by  means  of  a  phosphor  bronze 
wire  about  200  cm.  in  length.     The  axis  of  the  coil  is 
coincident  with  those  of  the  galvanometer  coils  already 
described.     The  method  of  using  it  will  be  given  in  the 
lecture  on  the  determination  of  H. 

The  four  coils  i,  4  and  2,  3  are  connected  with  a 
switchboard  of  massive  bronze,  shown  in  diagram  in 
Fig.  7,  by  means  of  which  any  coil  can  be  used  by  itself 
or  any  two  or  three,  or  all  four  in  series  (directly  or  dif- 
ferentially) or  in  multiple.  Thus  a  considerable  range 


ii 


/^^  *4*> 

t 

(UNIVERSITY 

V   ^A,°* 


of  sensitiveness  may  be  obtained.  The  time  required 
to  change  from  one  combination  to  another  is  that 
necessary  to  insert  and  withdraw  certain  plugs  and  to 
throw  the  switches. 

The  following  are  the  dimensions  of  the  six  coils  and 
the  constants  of  the  galvanometer  with  various  arrange- 
ments of  the  coils  when  the  strength  of  the  field  is 
H=  0.1710. 

DIMENSIONS. 


cm 


J 

Coil 

Measurements  uy  jxyau  aim  n 

i        radius        = 

aiiiiiiuii,  loot) 

100.1047 

tt 

4 

100  1275 

n 

2                   "..•,.== 

80.1037 

a 

3 

80.1025 

" 

W±  mean  radius  = 

76.0765 

u 

TFa      "          "       = 

76.0919 

Distance 

apart  (2  x)  i  to  4  ; 

99.9770 

« 

"       (2  x)  2  to  3  ; 

80.0274 

(2  x)W1toW2  76.0310 


FIG.    Q. 

CONSTANTS  (C.  G.  S.) 


ARRANGEMENT. 

G 

logGH 

*-f 

log  AT 

Coil  i 

80  06 

.044942 

—  2.652656 

3.8049 

0.580340 

Coils  i-f~4  

.089888 

—  2.053703 

I.9O24 

0.279293 

Coil  2 

"   3  

.056159 

—  2.749421 

3.0449 

0.483575 

Coils  2  -f  3  

.112319 

—  1.050450 

::  ^tfe: 

Wound  coils  in  series  

.202205 
.022430 

.729752 

—  1.305792 
—  2.350831 
—  i  863175 

0.84568 
7.6237 
0.23433 

—  1.927204 
0.882165 
—  1.369821 

12 


The  method  of  reading  the  deflections  of  this  gal- 
vanometer is  as  follows  :  There  are  two  circular  scales, 
graduated  decimally  upon  metal  strips.  These  strips 
form  two  opposite  quadrants  of  the  inner  face  of  a 
cylinder,  the  radius  of  which  is  50  cm.  (see  Fig.  8).  At 
the  centre  of  this  cylinder  is  the  mirror,  circular  in  form 
but  cut  in  two  diametrically,  the  halves  hinged  at  the 
median  line  and  dropped  down  to  an  angle  of  45°  with 
the  horizontal  plane,  to  which  position  they  are  adjusted 
by  means  of  screws  (see  Fig.  9). 

The  scale  is  viewed  through  a  telescope  (Fig.  10)  be- 
fore the  objective  of  which  is  placed  a  large  right- 


FIG.    10. 


angled  total-reflection  prism.  This  catches  the  vertical 
rays  proceeding  from  the  two  scales  to  the  mirror  and 
reflected  upwards  from  the  oblique  faces  of  the  latter. 
Images  of  those  portions  of  both  scales  which  are 
directly  opposite  the  halves  of  the  mirror  are  brought 
into  the  eye-piece  in  superposition.  These  may  be  read 
separately  by  cutting  out  one  or  the  other  by  means  of 
shutters. 

This  galvanometer  has  been  briefly  described  by 
Professor  W.  A.  Anthony,*  under  whose  direction  it 
was  constructed  in  1885. 

*  The  Electrical  Engineer  (N.  Y.),  Vol.  iv.,  October,  1885. 


13 


LECTURE  II. 

The  sine  galvanometer . — In  the  case  of  the  sine  galvano- 
meter ,  the  coils  of  which  are  free  to  revolve  upon  a 
vertical  axis,  and  are  made  to  follow  the  needle  in  its 
deflection,  the  formulae  of  the  tangent  galvanometer 
are  applicable  with  the  following  slight  modification. 

Since  the  deflected  needle  is  always  finally  in  the 
plane  of  the  coils  the  couple  due  to  the  current  acting 
upon  it  is  2  I  fi  instead  of  2  I  ff  cos  $,  and  equations  4 
and  5  become 

/'•=/,  sin  0.     •  (10) 

TT 

i  =  -@  sin  &.  (11) 

In  the  sine  galvanometer  the  angular  movement  of 
the  coils  may  be  indicated  by  verniers  upon  an  astrono- 
mical circle,  a  method  capable  of  greater  accuracy  than 
the  usual  methods  of  reading  the  deflection  of  a  tan- 
gent galvanometer.  The  limit  of  accuracy  is  deter- 
mined, however,  not  by  the  fineness  of  the  circle  in 
question,  but  upon  the  precision  with  which  the  coinci- 
dence of  the  needle  with  the  plane  of  the  coils,  in  the 
final  adjustment  of  the  latter  can  be  ascertained.  The 
most  refined  device  for  this  purpose  is  due  I  believe,  to 
Professor  H.  A.  Rowland.  It  consists  of  a  small  read- 
ing telescope  carried  upon  an  arm  which  turns  with  the 
coils  of  the  galvanometer.  This  telescope  bears  a  short 
horizontal  scale,  the  central  division  of  which  is  in  the 
same  vertical  plane  as  the  axis  of  the  telescope.  A 
mirror  attached  to  the  galvanometer  needle  gives  an 
image  of  the  scale  in  the  eye-piece  of  the  telescope,  the 
zero  falling  upon  the  cross  hair  when  the  plane  of  the 
coils  is  parallel  to  the  axis  of  the  needle. 

Standard  galvanometers  with  variable  constants. — It  is 
often  desirable  to  be  able  to  vary  the  constant  of  a 
standard  galvanometer  in  a  determinate  manner.  To 
vary  /Tin  such  a  manner  is  not  easily  practicable,  but  G 
may  be  subjected  to  perfectly  definite  changes.  The 
galvanometer  of  Obach*  affords  an  illustration  (Fig.  1 1 
is  from  Obach 's  original  plate)  of  one  method  of  accom- 
plishing this  end.  It  depends  upon  the  fact  that  the 

*  Obach :  Carl's  Repertorium  14,  p.  507,  1878. 

14 


constant  G  of  a  galvanometer,  with  needle  in  the  plane 
of  the  coil,  is  inversely  proportional  to  the  projection 
of  the  radius  of  the  coil  upon  the  vertical  north  and 
south  plane.  By  mounting  the  coil  of  a  tangent  gal- 
vanometer upon  a  horizontal  axis  and  causing  it  to 
make  any  angle  6  with  the  vertical,  the  effective  con- 
stant of  the  coil  can  be  given  any  value  G0  =  G  cos  0. 
The  equation  of  the  galvanometer  then  becomes 

H  II 


=  -T    tan      = 


GcosO 


tan  #. 


FIG.    II. 


Another  well-known  form  of  standard  instrument 
with  variable  constant,  is  the  Thomson  graded  galvano- 
meter, in  which  the  needle  is  moved  along  the  axis  of 
the  coil.  The  sensitiveness  of  instrument  is  thus  within 
definite  control  through  a  wide  range.  For  the  double 
purpose  of  further  increasing  the  range  of  usefulness 
and  of  protecting  the  needle  from  magnetic  disturbance 
the  galvanometer  is  provided  with  an  artificial  field.  A 
further  discussion  of  this  feature  will  be  given  in  the 
lecture  on  galvanometers  with  artificial  fields.  This 


type  of  instrument  like  the  "  swinging-arm  "  galvano- 
meter of  Moler  and  other  forms  does  not,  however,  be- 
long to  the  class  of  galvanometers  now  under  considera- 
tion, since  the  constant  is  determined  by  calibration  in- 
stead of  being  computed  directly  from  the  dimensions 
of  the  coil  and  the  position  of  the  needle. 

Corrections  to  be  applied  in  the  case  of  galvanometers  in 
which  the  conditions  already  described  are  not  fulfilled.  —  In 
the  development  of  the  law  of  the  galvanometer:  given 
in  lecture  i  ,  the  following  conditions  were  assumed  : 

1.  The  needle  is  in  the  axis  of  the  coils  ; 

2.  The  coils  are  vertical  and  in  the  magnetic  meri- 
dian ; 

3.  The  length  of  the  needle  is  small  as  compared  with 
the  radius  of  the  coils.     It  is  further  assumed  that  the 
number  of  turns  is  so  small  that  they  can  all  be  wound 
within  a  space  such  that  the  cross-section  of  the  bundle 
of  wires  is  small  as  compared  with  the  radius  of  the 
coil. 

The  theory  of  the  action  of  a  circular  current  upon  a 
needle  situated  at  a  distance  from  the  axis  is  given  by 
Maxwellf  also  by  Mascart  and  Joubert,^  and  has  been 
reproduced  in  many  other  treatises. 

It  will  be  possible  here  to  indicate  only  the  outlines 
of  the  analysis,  and  the  theoretical  considerations  upon 
which  the  selection  of  certain  forms,  notably  of  the 
Helmholtz  type  of  galvanometer  is  based. 

It  is  usual  to  base  this  discussion  upon  the  following 
considerations  : 

1.  For  any  point  situated  at  a  distance  y  from  the 
axis  of  circular  magnetic  layer  of  unit  density  the  po- 
tential p  is 

/>  =  2r(/0+/Iy!+/2y4+....etc.)        (12) 

in  which  f0  J\  /*8,  etc.,  are  all  functions  of  the  distance 
of  the  point  from  the  plane  of  the  sheet. 

2.  The  potential  at  the  same  point  due  to  a  magnetic 
shell  of  unit  strength,  the  boundary  of   which   is  the 
same     as    that    in    the    previous    case    is     V,    where 


3.  The  x  component  of  the  magnetic  force  due  to  a 
circular  current  traversing  the  boundary  of  the  mag- 
netic shell  is 

+  Maxwell  :  Electricity  and  Magnetism,  Vol.  II.,  Chap.  14,  also  in  Art.  711,  p.  355 
(Ed.,  1892). 

%  Mascart  et  Joubert  :  Lecons  sur  1'electricite  et  sur  le  magnetisme,  T.  2,  pp.  101, 
etc. 

16 


^rN 

UNIVERSITY) 


d  V 


The  quantities 
as  follows  : 


~     "  22  '   d  a? 


(14) 
o,  /i,  /2,  etc.,  are  related  to  each  other 


1           d2 

/„_,            i 

dafo 

Jn             2  ri*  '    d 
etc. 

'3?               ±  (2  .  4  .  . 

.  nf  d  a?n 

FIG.    12. 


The  value  of  /0,  however,  from  which  all  the  higher 
coefficients  are  readily  derived,  is  determined  by  the 
value  of  the  potential  upon  the  axis.  At  the  distance 
x  from  the  plane  of  the  layer  for  example  the  potential 


is — 


—  2  TT  ( 


—  a?)  ; 


(16) 


where  r  is  the  radius  of  the  circular  layer,  a  form  which 
leads  by  successive  differentiation  to  the  same  expres- 
sion for  the  magnetic  action  of  the  current  as  that  given 
in  lecture  I.  (equation  3).  Thus  : 


d  x  ~ 


dx 


r2)! 


(18) 


which  is  identical  with  equation  3  when  the  current  is 
unity. 

Since,  however,  p0  is  a  particular  value  of  p  (equation 
12),  in  which  y  =  0  we  have 


/0  =  AJM^  —  «,  -  «  -  *  y  (19) 

where  -w2  =  r2  -|-  ar2. 

The  coefficients  belonging  to  the  series  for  X  may 
readily  be  obtained  by  differentiation. 

Thus  we  have 

(20) 


/o  =  u  —  x, 

_r  \.  CL 


/O 


=  -<?  /8,etc.; 


c   'W       a? 
aJ  a?  ~~  ^ 


d  a?  ~  u5 

ft  u      3  y2  (4  a?  —  r2) 

<#  a?4  ~  'w7 

By  means  of  these  values  we  may  write  the  series  for 
X  to  the  fourth  power  of  x. 

3    4^  —  ^ 


32..  5  y*  —  12  r2  a?2  -f  8  a?4 


The  series,  the  lower  members  of  which  are  given  in 
22  leads  to  a  result  only  when  y  <  u,  which  is  always 
the  case  in  dealing  with  galvanometers  of  ordinary 
form. 

When  the  needle  lies  in  the  axis,  y  becomes  zero,  and 
equation  22  becomes 

T-_    f       27rr*          2*7* 

-/*•-    ^3    -pqr^; 

which  is  identical,  of  course,  with  18  and  with  3. 

Professor  W.  S.  Franklin  has  suggested  to  the  writer 
that  an  expression  for  X  corresponding  to  22  may  be 
obtained  without  recourse  to  the  artificial  conception  of 
the  potential  of  a  magnetic  shell, 

18 


His  method  is  as  follows  : 

To  find  a  development  of  the  component  in  the  direction 
of  the  axis  of  the  magnetic  field  due  to  a  circular  coil. 

The  value  of  this  component  at  points  in  the  axis  (at 
these  points  the  component  being  the  total  field)  is  ob- 
tained as  follows  : 

Let  a  magnetic  pole  of  strength  m  be  placed  in  the 
axis  of  the  coil  at  a  distance  x  from  its  plane.  The 

m         m 

magnetic  field  /  at  the  wire  is  /  =  ^  =y2  .  ^.  An  ele- 
ment d  I  of  the  coil  will  be  acted  upon  by  a  force  d  F 
at  right  angles  to  /  and  to  d  I,  such  that  d  F  =  f  i  d  I. 
i  being  the  strength  of  the  current  in  the  coil.  The 
component  of  d  Fin  the  axial  direction  is  /  i  d  I  Sin  0, 
and  this  same  force  acting  upon  every  element  d  I  of 
the  coil  gives  for  the  total  force  acting  on  the  coil 
(which  force  is  in  the  axial  direction  from  symmetry) 
>  =  /  Hsin  0. 

T 

where  I  =  2  n  T  and  sin  6  — 


FIG.  13. 


Substituting    in     this    value    for    F    the    values    of 
f  •=.   a.*,  I,  and  cos  6  we  have 


(23) 


=  2  TT  m  i  a 


+  flty 


Owing  to  the  equality  of  action  and  reaction  this 
same  force  must  act  upon  the  pole  m,  but  a  force  acting 
on  a  magnetic  pole  is  always  the  product  of  the  strength 
of  the  pole  into  the  strength  of  the  magnetic  field  at  the  pole 
so  that  the  factor  by  which  m  is  multiplied  in  (i)  is  the 
required  strength  X  of  the  magnetic  field  due  to  the 


circular  coil  at  a  distance  from  x  the  plane  of  the  coil 
and  in  its  axis  so  that 


2  7i  i 
r 


in  which  expression  , _* may  easily  be  expanded 


in  powers  of    ?!_,  giving  a  series  for   -5T  with   definite 
r 

coefficients. 

The  component  in  any  given  direction  of  a  magnetic 
field  satisfies  La  Place's  equation  so  that  we  may  ex- 
pand such  a  component  when  symmetrical  to  an  axis  in 
a  series  of  zonal  harmonics,  i.e., 

X=AQPv  (cos  ^)+A1dPz  (cos0+A4A(cos^).  . .  (25) 

where  the  ^.'s  are  undetermined  coefficients  X  is  the 
required  component  at  distance  d  from  centre  of  circular 
coil  d  making  an  angle  <p  with  the  axis  of  the  coil  (see 
Fig.  13).  When  <p  =  0  then  (25)  must  reduce  to  the 
value  (24)  for  JTat  points  in  the  axis.  Substituting  in 

1 

(24)  the  expansion  of  /         cc2\f  and  placing  this  series 


equal  to  (25)  with  <p  =  0  we  have  (since  Pn  (cos  <p)  —  1 
when  <p  —  0  and  d  becomes  x  when  <p  =  0),  by  placing 
the  various  coefficients  of  x,  x*,  x3,  etc.,  equal  each  to 
each  a  means  for  determining  the  values  of  the  A's  so 
that  (25)  is  then  the  completely  determined. 
The  series  for  JTis  made  up  thus  : 

PQ   (COS    <p)    —     1 

Pl  (cos  (p)  —  cos  (p 

P2  (cos  f)  =  i  (3  cos2  <p  —  1)  (26) 

P3  (cos  tp)  =  J  (5  cos3  ^  —  3  cos  ^  ),  etc. 

The  development  thus  formed  holds  only  when  x  <  r. 
For  values  of  a;  >  r  must  be  broken  up  so  as  to  intro- 


duce  the  development  of  /r3          \|  when  we  may  pro- 

V?+ 

ceed  as  before. 

20 


The  other  rectangular  components  of  the  magnetic 
field  due  to  a  circular  coil  cannot  be  developed  in  a 
series  of  zonal  spherical  harmonics. 

The  value  of  the  series  (22)  lies  in  the  fact  that  by 
means  of  it  we  may  discover  the  influence  of  deviations 
of  the  galvanometer  needle  from  the  axis  when  the 
needle  is  sitiiated  at  various  distances  from  the  plane 
of  the  coils.  This  has  been  done  by  a  graphical  method 
in  Mascart  and  Joubert's  treatise  (Vol.  II.,  p.  103). 
Curves  showing  the  sign  and  value  of  the  two  members 
containing  y*  and  y*  respectively,  show  that  the  former 
which  is  the  more  important  becomes  zero  when  x  =  %  r 
while  the  member  containing  y*  is  very  small  for  that 
value  of  x.  It  is  in  accordance  with  the  results  of  this 
analysis,  therefore,  that  in  the  Gaugain  galvanometer 
and  in  Helmholtz's  pattern  also  the  distance  from  the 
needle  to  the  plane  of  the  coils  is  always  one  half  the 
radius  of  the  latter. 

It  is  not  always  allowable  in  computing  the  constant 
of  a  galvanometer,  to  take  a  mean  radius  r  as  applicable 
to  all  the  turns  which  the  coil  contains.  Particularly  in 
the  case  of  galvanometers  with  many  turns  of  the  wire 
it  becomes  necessary  to  consider  the  cross-section  of 
the  coil.  For  the  general  discussion  of  this  case  the 
reader  is  referred  to  Maxwell's  treatise.* 

Correction  for  the.  length  of  the  needle.  —  The  influence  of 
the  length  of  the  needle  is  a  subject  which  scarcely 
needs  attention  when  dealing  with  modern  galvano- 
meters of  the  type  now  under  consideration.  The  cor- 
rection is  a  small  one  in  all  ordinary  cases.  It  is  a  mini- 
mum in  the  Helmholtz  galvanometer  in  which  the  dis- 
tance from  the  plane  of  the  coil  to  the  needle  is  -£  r. 
In  instruments  of  this  type  the  correction  is  as  follows, 
where  length  of  needle  is  2  I. 

2  I  =  .  2  r  ;  correction  .  001 

2  £  =  .1667-;         "        .0005,  etc. 


These  lengths  are  much  greater  than  any  in  use  in 
modern  tangent  galvanometers,  in  the  case  of  which 
instruments  the  correction  becomes  entirely  neglible. 

Correction  for  torsion.  —  In  tangent  galvanometers  the 
correction  for  the  torsion  of  the  suspension  fibre  is  so 
small  that  the  following  simple  approximate  method  of 
determining  it  is  entirely  adequate. 

To  estimate  the  correction  for  torsion  we  twist  the 

*  Maxwell  :  A  Treatise  on  Electricity  and  Magnetism,  Vol.  II.,  Chap,.  15  p.  354 
(edition,  1892). 

21 


upper  end  of  the  suspension  fibre  through  an  angle  8. 
and  note  the  movement  of  the  needle  resulting  there- 
from. Let  the  angle  be  ft  ;  then 

(27) 
u 

is  the  approximate  factor. 

Case  in  which  the  coils  of  the  galvanometers  are  not  in  the 
magnetic  meridian. — Let  the  coils  make  an  angle  a.  with 
the  meridian.  Then  upon  reversing  the  direction  of  the 
current  through  the  instrument  we  get  equilibrium  for 
the  following  positions  of  the  needle. 

TT 

i  cos  (#  +  a)  =  77  sin  #.  (28) 

TT 

i  cos  (01  —  a)  =  -Q  sin  tf1.  (29) 

Adding  these  equations  we  have 

i  [cos  &  cos  a  —  sin  $  sina  -f-  cos  $'  cos  a  -|-  sin  #'  sin  «  = 

^(sin  #  +  sin  #')         (30) 
from  which  by  the  use  of  the  usual  conversion  formulas 

•f  #  — #'    i      -       sin  #  —  #'-] 

*  [cos  a  cos g-   +  sin  a g-  (31) 

H         &  +  #'          #  —  #' 
0.tan»2—  -cos -3- 

When  a  is  very  small,  the  usual  case,  the  member 

cit"|    7jr   ^-XTI— i    7/1 

sin  a  -    — ^ —    -  which  is  the  product  of    two  small 

quantities  disappears  and  cos  a  is  nearly  unit.  We  may 
use  then  as  an  approximate  form 


When  a  is  not  small  the  complete  expression  must  be 
obtained  as  follows  : 

From  28  and  29  we  have, 

H    sin# 
cos  #  cos  a  —  sm  #  sm  a  —  -/•> j —  and 

77     sin  #' .     jj 

cos  #  cos  a-j-  sin  #  sm  a  =  75  •  — : —  >  ais° 


22 


2  H  sin  &  sin 

cos  a  = 


a  s 


*        sn 

,      2  jy  sin  #  sin  #' 
COS 


,   {  2  jy  sin  #  sin  #'  ) 
<M  a  £  sin  (#+#'))    "" 


.      n   A/         4r^T8sin  ^sin2^         ^sin  & 
"^^sin2(^  +  ^)  r         ^^ 

(33) 
When  solved  for  $  this  becomes 

^  _  #2  jsin  ^  cos  ^  —  sin  $  cos  #7  +  sin2  &  sin2  #xj 
(?2  (sin  #  cos  ^  +  cos  #  sin  ^)2 

which  may  be  written  in  the  following  more  convenient 
form 

,  _  jH*  (tan  &'  —  tan  #)2  +  4  tan2  #  tan2  &'      ,„.. 
G*  (tan  &  +  tan  #')8 

Since  in  nearly  all  cases  the  galvanometer  is  placed 
approximately  with  its  coils  in  the  magnetic  meridian, 
the  simpler  expression  (32)  is  of  sufficient  accuracy. 


LECTURE   III. 

Ballistic  Methods, — It  is  frequently  desirable  to  deter- 
mine current  values  by  means  of  a  single  "  throw  "  or 
"  kick  "  of  the  galvanometer  needle,  instead  of  by  the 
method  of  permanent  deflections.  In  the  measurement 
of  induced  currents,  or  of  transient  or  rapidly  fluctuat- 
ing current  of  any  kind  this  method  is  almost  the  only 
one. 

A  galvanometer  thus  used,  an  instrument,  that  is  to 
say,  which  is  arranged  for  measuring  the  time-integral 
Q  or  total  charge  transmitted  during  the  flow  of  an 
electric  current  of  short  duration,  is  called  a  ballistic 
galvanometer,  but  the  term  should  be  applied  to  the 
method  rather  than  to  the  instrument  since  any  galva- 
nometer may  be  used  ballistically.  It  is  true,  however, 
that  the  method  demands  certain  qualities  which  are 
not  essential  nor  desirable  in  instruments  intended  for 
use  in  the  method  of  permanent  deflections.  Thus 
there  has  arisen  a  type  of  galvanometer  especially  de- 
signed for  the  method  of  the  first  throw. 

An  expression  for  the  performance  of  a  galvanometer 
when  used  ballistically  may  be  derived  in  the  following 
manner.  The  force  (//)  (see  Lecture  I.)  due  to  a  cur- 
rent i  flowing  through  the  galvanometer  coils,  the  con- 
stant of  which  is  6r,  is 

/,  =  G  i  m,  (35) 

and  the  moment  of  the  couple  at  the  needle  pole  is 
GiZlm  =  GiM. 

This  couple  acting  during  the  time  £,  upon  a  needle 
the  moment  of  inertia  of  which  is  7",  will  produce  an  an- 
gular velocity 

(36) 


In  the  case  of  galvanometers  used  ballistically,  it  is 
not  with  the  current  itself  but  with  the  time-integral 

( Q  =  /  i  d  t)  that  we  are  concerned,  and  equation  (36) 
becomes 

, .  &  -M-  *£  (^T\ 

From  this  equation,  G,  M  and  I  being  known,  Q 
could  be  obtained ;  provided  that  the  angular  velocity 
could  be  directly  observed. 

A  slightly  different  statement  of  this  point  is  the 
following  : 

A  current  i  in  the  coils  of  the  galvanometer,  which  is 
supposed  to  be  placed  in  the  earth's  field  or  in  an  arti- 
ficial field  the  strength  of  which  is  jET,  the  plane  of  the 
coils  being  in  the  magnetic  meridian,  produces  a  field 
perpendicular  to  H.  The  strength  of  this  field  at  the 
centre  of  the  coil  is 

F=  Gi  (38) 

where  6r,  as  in  previous  equations,  is  the  constant  of 
the  coils.  Let  to  be  the  angular  velocity  of  the  sus- 
pended parts  of  the  galvanometer  at  a  given  instant  of 
time.  If  any  torque  $  act  upon  the  moving  system,  to 
will  change  in  such  a  manner  that 

(39) 

Now  the  field  F,  being  perpendicular  to  the  axis  of 
the  needle  exerts  upon  it  a  torque  ®,  which  is  equal  to 

FM,  or  to  G  i  M  (38).     Since,  however,  i  =  -=-^, 

a  t 

we  have 

/^L-f?  —  G  M  ~z  (40) 

dt  dt 

If  to  is  zero  at  the  instant  of  closing  the  galvano- 
meter circuit,  equation  (40)  becomes 

I  to  =  GMQ,  (41) 

which  is  the  same  as  (37),  the  fundamental  equation  of 
the  ballistic  galvanometer. 

In  order  to  render  the  ballistic  method  feasible,  it  is 
necessary  to  avoid  the  determination  of  the  moment  of 
inertia  (I)  and  the  magnetic  moment  ( M )  of  the  needle, 
also  the  impracticable  operation  of  observing  the  angu- 

25 


lar  velocity  (to)  ;  and  to  substitute  the  easily  measured 
time  of  vibration  T  and  the  observation  of  the  ampli- 
tude of  the  first  throw  ($.)  That  it  is  possible  to  do  so 
will  appear  at  once  from  the  consideration  of  the  work 
done  upon  the  needle,  to  deflect  it  through  the  angle  & 
of  the  first  throw.  To  move  the  north -pointing  pole  N 
(Fig.  14)  the  strength  of  which  is  m,  from  a^  to  b,  and 
the  south-pointing  pole  S,  of  like  strength,  through  a 
similar  path,  requires  work  against  the  force  Jim. 
This  amounts  to 


2  H  m  X  di<h  —  2  H  M  (1  —  cos  #) 

where  M  is  the  magnetic  moment  of  the  needle. 

Expressed  in  terms  of  the  moment  of  inertia  (7)  and 
the  angular  velocity  (co)  of  the  returning  needle,  at  the 
instant  when  the  angle  #  becomes  zero,  the  equivalent 


FIG.  14. 
of  this  work  is  1  1  to2  ,  whence 


to  =  2  .  sin  !  #  (42) 

/  2 

By  combining  equations  37  and  42  we  obtain 

«  -  w^W7™  \  »  =  '-r^m*  ¥>  (43) 

and  by  use  of  the  expression  for  the  time  of  vibration 

of  the  needle,  T  =  n  V  -J—  (44) 

H.  M. 

we  may  reduce  the  equation  of  the  ballistic  galvano- 
meter to  the  final  form, 


L#.  (45) 

26 


Absolute  measurements  with  the  ballistic  galvanometer 
involve  :  — 

1.  A  knowledge  of  the  constant  (r,  to  be  determined 
by  measurement. 

2.  A  knowledge  of  Hio  be  obtained  by  one  of  the 
methods  to  be  described  in  Lecture  IV. 

3.  A  knowledge  of  ^in  seconds. 

4.  The  observation  of  the  throw  $. 

The  value  of  77is  that  for  infinitesimal  amplitudes, 
which  is  to  be  obtained  by  the  application  of  a  correc- 
tion similar  to  that  used  in  determining  the  rate  of 
vibration  of  a  magnetometer  needle.  (See  Lecture  IV.) 

The  separate  determination  of  H  and  G,  which  latter 
quantity  can  not  be  satisfactorily  ascertained  from 
measurements  of  the  coils  of  the  forms  of  galvano- 
meter usually  employed  in  ballistic  work,  may  be 
avoided  by  calibration  of  the  instrument.  If  a  known 
current  ip  be  sent  through  the  galvanometer  and  the 
permanent  deflection  <p  be  noted,  we  have 

(46) 


. 

a       tan  <p 

and  the  expression  for  Q  becomes 

Q=     **'     .  rsinlfl.  (47) 

TT  tan  <p  .  2 

Equation  (47)  affords  an  expression  for  $in  which  all 
the  factors  are  either  numerical  or  capable  of  computa- 
tion from  direct  observation.  If  the  value  of  Q  is  to 
be  free  from  errors  other  than  those  involved  in  the 
determination  of  ipy  T,  <p,  and  #  however,  the  condi- 
tions implied  in  the  equations  upon  which  the  expres- 
sion (47)  is  based  must  be  complied  with.  These  condi- 
tions may  be  stated  as  follows  : 

Conditions  implied  by  equation  41  :  — 

1.  The  needle  must  be   at  rest  when  the  discharge 
through  the  coils  begins. 

2.  The  coils  must  be  in  the  plane  of  the  directing 
field  (H.) 

3.  The  field  (F)  due  to  the  current  in  the  coils  must 
remain  sensibly  perpendicular  to  the  axis  of  the  needle 
during  the  entire  discharge,  so  that  the  torque  may  be 
equal  to  G-  i  M  through  the  whole   time  of   its  action. 
This  condition  compels  a  slow  vibration  period  of  the 
needle  excepting  in  cases  in  which  the  duration  of  the 
discharge  is  very  brief. 

27 


Conditions  implied  by  equation  42  : — 

i.  The  whole  of  the  kinetic  energy  (•£  I  aF)  must  be 
employed  in  turning  the  needle  against  the  directing 
field.  This  means  that  there  must  be  no  damping,  a 
condition  which  can  not  be  fulfilled  in  practice,  and  for 
the  failure  to  fulfil  which  proper  corrections  must  be 
applied. 

Conditions  implied  by  equation  44  : — 

i.  The  suspending  fibre  must  be  free  from  torsion. 


FIG.    15. 

2.  The  damping  must  be  so  slight  as  to  produce  no 
sensible  influence  upon  the  time  of  vibration  of  the 
needle. 

Condition  implied  by  equation  46  : — 

i.  The  diameter  of  the  coils  must  be  large  compared 

28 


with  the  length  of  the  needle  ;  in  other  words,  the  law 
of  the  tangent  galvanometer  must  hold  true  for  such 
deflections  as  are  produced  by  the  current  used  in  cali- 
brating the  instrument. 

Many  of  these  conditions  coincide  with  those  desir- 
able in  the  magnetometer  and  accordingly  the  earlier 
forms  of  ballistic  galvanometer,  of  which  Fig.  15  affords 
an  illustration,  were  simply  magnetometer  bars  with 
one  or  more  elongated  coils  of  wire  surrounding  them. 
The  requirements  of  modern  practice  and  the  extension 
of  the  ballistic  method  to  operations  of  the  highest 
delicacy  have  essentially  modified  the  type. 

The  equation  of  the  ballistic  galvanometer  already 
given  (Equation  37)  involves  the  throw  of  an  undamp- 
ed needle.  In  the  case  of  a  needle  moving  through 
a  resisting  medium  (damped  by  friction)  or  in  the  neigh- 
borhood of  metallic  bodies  within  which  induced  cur- 


FIG.  16. 


rents  are  generated  by  the  moving  field  of  the  needle 
(damped  by  induction)  the  deflection  #  will  always  be 
less  than  that  which  would  be  observed  in  the  case  of 
an  undamped  needle,  by  an  amount  which  may  be 
estimated  as  follows  : 

Determination  of  the  effect  of  damping  : — Damping  affects 
the  values  of  T  and  of  $.  To  determine  its  influence 
on  these  quantities  we  take  advantage  of  the  following 
law  : 

Law  of  logarithmic  decrements: —  The  amplitudes  of  succes- 
sive oscillations  form  a  decreasing  geometric  series  and  the 
natural  logarithms  of  successive  swings  have  a  constant 
difference.  This  difference  is  known  as  the  logarithmic 
decrement. 

29 


The  relation  of  the  damped  to  undamped  vibra- 
tions of  the  same  period  may  be  shown  graphically  by 
deriving  the  curve  for  the  former  by  a  method  analogous 
to  that  in  which  the  curve  of  sines  is  derived. 

Tait*  has  shown  that  the  logarithmic  spiral  (Fig  16) 
gives  the  law  of  damped  vibrations  in  the  same  way 
that  the  circle  leads  to  the  curve  of  sines.  We  derive 


FIG.    17. 

the  latter  curve  by  considering  the  motion  of  a  radius 
of  the  circle  traveling  with  a  uniform  angular  velocity 
and  plotting  a  curve  (Fig.  17)  with  times  as  abscissae 
and  the  corresponding  vertical  distances  of  the  moving 
end  of  the  radius  as  ordinates.  A  similar  procedure 
applied  to  the  case  of  the  radius  vector  of  the  spiral 
(Fig.  1 8)  gives  us  the  curve  of  damped  vibrations. 


FIG. 


In  order  that  the  decrement  shall  be  strictly  logarith- 
mic the  resistance  offered  to  the  needle  must  be  pro- 
portional to  the  velocity.  Atmospheric  resistance  is 
thus  proportional  for  low  velocities  and  for  high  ones  it 
deviates  from  that  law  and  becomes  more  and  more 
nearly  proportional  to  the  square  of  the  velocity.  For 

*Tait ;    Proceedings  of    the    Royal   Society  of    Edinburgh,    1867 ;    also    Maxwell's 
Treatise,  vol.  ii.,  p.  375. 

30 


all  velocities  reached  by  galvanometer  needles,  however, 
the  former  law  may  be  assumed.  Even  in  the  case  of 
instruments  of  the  D' Arson  val  type  the  divergence  is 
not  marked.  Fig.  19  is  from  the  photographic  trace  of 
a  very  small  light  mirror  and  needle  mounted  in  a 
strong  field  for  the  purpose  of  following  the  changes  in 
rapidly  fluctuating  currents.  [Messrs.  Hotchkiss  and 


FIG.  19. 

Millis,  1893.]  It  shows  the  sudden  throw  av  b±  upon 
making  circuit  and  the  decrement  of  the  oscillations  as 
the  needle  comes  gradually  to  rest  in  the  new  position 
at  c.  The  period  was  about  IODO  vibrations  per  second. 
The  record  was  obtained  by  shooting  a  sensitive  photo- 
graphic plate  rapidly  across  the  field  of  a  camera  and 


t    \  N 


FIG.    2O. 


developing  the  image  of  the  spot  of  light  thrown  upon 
the  same  from  the  vibrating  mirror. 

Working  formula  and  approximate  correction  for  damping: — 
For  work  requiring  no  very  great  accuracy  equation 
(47)  may  be  greatly  simplified.  Let  a  (Fig.  20)  be  the 
observed  throw,  in  scale  divisions,  produced  by  the  dis- 


charge.  Let  b  be  the  permanent  deflection,  in  scale 
divisions  produced  by  the  current  ip  and  let  D  be  the 
distance  of  the  scale  from  the  mirror.  Then  we  have 

approximately  Sin  \  &  =  _^—  and  tan  <p  =  —  _  whence 

4  _Ls  2i  dJ 

Equation  (47)  becomes. 

-      -       (48) 


When  a  vibrating  body  has  reached  its  extreme  posi- 
tion on  either  side  of  its  position  of  equilibrium  it  is 
said  to  in  elongation. 

The  zero  point  (using  telescope  and  scale)  is  the  scale 
reading  when  the  body  is  in  its  position  of  equilibrium. 
The  zero  point  is  most  easily  determined  by  observing 
an  odd  number  of  successive  elongations,  an  even  num- 
ber to  the  right  and  an  odd  number  to  the  left.  The 
mean  of  the  right  hand  readings  and  the  mean  of  the 
left  hand  readings  are  taken.  The  mean  of  these  two 
gives  the  zero  point.  When  the  damping  is  great  this 
gives  a  distinctly  erroneous  value  for  the  zero  point. 
In  such  a  case  it  is  better  to  allow  the  vibrations  to 
cease  so  that  the  zero  point  may  be  read  off  directly. 

The  distance  (or  angle)  from  the  position  of  equili- 
brium to  an  elongation  is  called  also  an  elongation.  No 
confusion  need  arise  from  this  double  use  of  the  word 
elongation. 

The  successive  elongations  of  a  damped  vibrating 
body  as  has  already  been  shown,  form  ordinarily  a  de- 
creasing geometric  series,  in  which  if  k  be  the  ratio  of 
an  elongation  to  the  next  following, 


am 


(49) 


Wherein  «0  is  an  elongation  and  a^  is  the  mth  follow- 
ing elongation,  loth  of  which  are  easily  observed.  The 
symbol  k  is  the  ratio  of  damping. 

The  observed  throw,  a,  in  equation  (48)  is  smaller  in 
the  ratio,  nearly,  of  i  :  V  k  than  it  would  be  were  there 
no  damping  so  that  we  may  write  \'  k-  a  for  a  in  that 
equation,  whence  : 

(50) 


TC.l 

If  k  is  very  nearly  unity  this  equation  leaves   only 
very  small  outstanding  reduction  errors. 

32 


There  are  three  distinct  kinds  of  error  which  affect 
the  results  of  a  set  of  observations,  (a)  Observational 
errors,  (b)  Instrumental  errors,  (c)  Reduction  errors. 
Errors  of  observation  arise  from  incorrect  determina- 
tions of  the  immediately  observed  quantities  ;  Instru- 
mental errors  arise  from  incomplete  realization  of  the 
conditions  assumed  in  the  derivation  of  the  formulae 
for  reduction  ;  Reduction  errors  arise  from  the  use  of 
approximate  reduction  formulae,  often  indeed  from 
the  use  of  approximate  formulae  in  the  allowance  for 
instrumental  error. 

Example  : — Errors  in  the  observed  values  of  <p,  & 
and  Tin  equation  (47)  lead  to  observational  errors  in  Q. 
Any  failure  in  the  realization  of  the  conditions  assumed 
in  the  derivation  of  (47),  if  ignored,  leads  to  an  instru- 
mental error  in  Q.  The  use  of  approximate  formulae, 
such  as  (48),  and  (50)  in  the  calculation  of  Q  leads  to 
reduction  errors  in  Q. 

When  great  accuracy  is  desired  allowance  should  be 
made,  if  possible,  for  all  instrumental  error,  and  approxi- 
mate formulae  should  not  be  used  unless  the  reduction 
errors  they  introduce  are  decidedly  smaller  than  the 
observational  error  and  the  outstanding  instrumental 
error.  Instrumental  errors  are  so  called  from  the  fact 
that  the  conditions  assumed  in  the  derivation  of  reduc- 
tion formulae  refer  to  the  arrangement  and  construction 
of  the  apparatus  used  in  taking  the  observations.  Cal- 
culations, which  are  made  in  the  allowance  for  instru- 
mental error,  are  always  based  upon  observed  values  of 
such  quantities  as  characterize  the  incompleteness  in 
the  realization  of  the  assumed  instrumental  conditions. 

Rigorous  correction  for  damping. — Every  actual  case  of 
vibration  is  damped.  This  damping  is  due  to  resisting 
forces  which  oppose  the  motion  of  the  vibrating  body. 
It  is  assumed  in  the  following  discussion  that  the  resist- 
ing forces  are  strictly  proportional  to  the  velocity  of 
the  vibrating  body.  The  errors  left  outstanding  by 
this  assumption  can  always  be  made  inconsiderable  by 
arranging  that  the  vibrating  body  be  massive  ;  that  it 
vibrate  slowly ;  and,  if  it  vibrate  about  an  axis,  that  it 
be  circularly  symmetrical  about  that  axis  and  smooth 
so  as  to  stir  the  air  as  little  as  possible.  With  such 
limitations  the  following  discussion  leads  to  rigorous 
correction  for  damping  in  the  use  of  the  ballistic  galva- 
nometer. 

Let  (p  be  the  angle  measured  from  the  position  of 
equilibrium  of  a  vibrating  body  to  its  instantaneous 

position.     Then,  ordinarily,  a  torque  1&  =  —  c  (f>  —  f— ~ 

€(>     t 


33 


acts  upon  the  body  and  since  this  torque  must  be  equal 

d2  d> 
to  1  -  —  I.  ,  /  being  the  moment  of  inertia  of  the  body 

Ct  11 

and  c  and  /  being  constant,  the  only  condition  which 
must,  in  general,  be  satisfied  by  the  angle  (j>  is  ex- 
pressed by  the  differential  equation 


in  which  2  /9  is  written  for  !—  and  f  for  —  . 

Equation  (51)  leads,  for  our  present  purpose,  to  the 
following  expression  for  the  instantaneous  value  of  the 
angle  <p  :  — 

<p  =  A  e~^  t  Sin  to  t  (52) 

in  which  A  is  an  undetermined  constant, 

o)  =    yr  __0a  (53) 

and  t  is  the  elapsed  time  reckoned  from  the  instant  of 
one  of  the  passages  of  the  body  through  its  position  of 
equilibrium 

The  time  interval,  T,  which  elapses  between  succes- 
sive null  values  of  <p  is  called  the  period  of  the  vibra- 

tion and  is  evidently  equal  to  —  or 

CO 

<»=    ^T  (54) 

Let  T'  be  the  undamped  time  of  vibration.  From 
(53)  and  (54)  with  the  .  condition  /3  —  o  we  have 

T'  =  -2u  ,  whence 

Vr  ' 


The  successive  elongations  are  maxima  and  minima 
values  for  (p  and  the  values  of  t  at  the  instants  of  these 

elongations  are  to  be  found  from  (  V    —  o  .     Applying 

a  t 
this  method  to  equation  (52)  we  easily  find 

t  =  —  Arc  tan  -^-  +  T  n  ,  (56) 

co  p 

in  which  n  is  any  whole  number.     [At  the  first  elonga- 

34 


tion  n  —  o  ,  at  the  second  n  =  i,  etc.]  Substituting  the 
value  of  t  from  (56)  in  equation  (52),  we  have  for  the 
first  elongation  <p  ' : — 

*>:=..  -g_  ;^«~"Arctant       (57) 


For  the  second  elongation  <p  ",  —  we  have  similarly 


also  <!>"'  =  e~       <!>" 

&c.  &c. 

The  ratio  of   damping,  k,  however,  is  by   definition 
equal  to  ^  whence 

k  =  e  ft  T  ,  (58) 

or 

;,  =  Log.  nat.  k  =  ft  T.  (59) 

This  quantity  ft  T  ,  which  for  convenience  will  here- 

after be  represented  by  A  is  the  logarithmic  decrement  of 

the   vibrations.      Expressed  in  terms  of  the  common 

system  of  logarithms  it  is, 

/  =  2.306  Log  10  k  . 
Noting  the  relations  between  ^  ft,  71  and  w  ;  viz  :  — 


we  may  evidently  write  equation  (55)  in  the  form 

T  '  =  T  -    J-  (60) 


The  value  of  (p  '  given  in  equation  (57)  is  the  observed 
throw  of  the  ballistic  galvanometer.  If  there  had  been 
no  damping  the  throw  would  have  been  A,  since  ^/  re- 
duces to  A  when  ft  =  o.  Solving  equation  (57)  for  A 
and  eliminating  ft  and  to  by  use  of  (54)  and  (59)  we 
have 

?.  7T 

TO"         -  arc  tan  -r- 
+  ^.e"  {  -f        (61) 

7T 

or 

35 


1  7T 

/ —      — ;-*  —  arc   tan  -^ 

A  =  V  1  +  ^  .  k  *  •  <!>'        (62) 

Equation  (61)  or  (62)  enables  us  to  calculate  the  un- 
damped throw  (A)  from  the  observed  throw  (^'). 

The  calculation  of  Q,  having  observed  T,  k,  y>,  #,  i p 
is  as  follows  : — 

From  the  observed  value  of  k,  equation  (49),  X  is  cal- 
culated. From  the  observed  period  T  the  undamped 
period  T'  [Same  as  t  in  equation  (47)]  is  calculated  by 
means  equation  (60).  From  the  observed  throw  y '  the 
undamped  throw  A  is  calculated  by  equation  (62). 
Then  T'  is  substituted  for  t  and  A  for  #  in  equation  (47) 
from  which  Q  is  calculated.  The  steps  in  this  calcula- 
tion are  tedious  and  cannot  be  greatly  simplified  unless 
approximate  formulae  are  used. 

A  formula  somewhat  more  exact  than  (50)  is  obtained 
as  follows  : — 

T 

Write  -7=     ==  for  T  in  (48) 

V  1  H 
[See  equation  (60)]. 

1  7T 

/ —      — J-ST         —  arc  tan  — 
Write  V  1  +  —  .  k  n  aior  am         (48) 

[See  equation  (62)]. 
We  thus  have 

1  7T 

</   T  n  Tf   ~  arc  tan^~ 
Q  _  ^•2-a'fc    *  l  9  (63) 

in  "which  Tis  the  observed  period,  a  is  the  observed 
throw,  etc.  The  exponent  of  k  in  (63)  becomes  ^  when 
k  is  very  nearly  unity  or  A  very  small  ;  so  that  for  this 
case  (63)  approximates  to  equations  (50). 


LECTURE  IV. 

Methods  of  Measuring  the  Magnetic  Field.  —  Since,  in 
the  use  of  the  galvanometer,  the  sensitiveness  depends 
upon  the  value  of  Hy  that  is  to  say,  upon  the  strength 
of  the  earth's  field,  or  of  the  artifically  created  field 
within  which  the  needle  swings,  it  is  necessary  to  be 
able  to  determine  the  strength  of  'this  field  with  a  high 
degree  of  accuracy. 

Two  of  the  earliest  methods  for  the  absolute  deter- 
mination of  H  were  those  developed  by  Gauss  and  by 
his  co-worker  in  Gb'ttingen,  Wilhelm  Weber.  Gauss' 
method  comprises  two  operations: 

(i.)  The  determination  of  the  rate  of  vibration  of  a 
suspended  bar  magnet. 

(2.)  The  observation  of  the  deflection  which  this 
magnet  is  capable  of  producing  when  acting  upon  an- 
other suspended  magnet  at  a  known  distance. 

From  the  time  of  vibration  we  get 


Where  Tis  the  time  of  vibration,  JTthe  moment  of 
inertia  of  the  magnet  and  m  the  strength  of  the  mag- 
net pole. 

From  the  determination  of  the  deflection  we  get 


772. 

-     =     ds  sin 


or  --—          tan 


(65) 


The  latter  expression  applies  when  the  deflecting 
magnet  is  stationary,  in  a  position  due  east  or  west  of 
the  magnetometer,  the  former  is  used  when  the  deflect- 
ing magnet  is  mounted  on  a  swinging  arm  as  in  the 
Kew  magnetometer. 

Equations  (64)  and  (65)  each  contain  m  and  H.  The 
quantity  m,  therefore,  may  be  eliminated,  and  H  may 
be  obtained  in  terms  which  involve  only  the  funda- 
mental quantities  —  length,  mass  and  time. 

37 


There  are  few  operations  in  experimental  physics 
which  can  be  carried  out  with  a  higher  degree  of  pre- 
cision than  this  first  operation  of  Gauss,  which  consists 
in  the  determination  of  the  time  of  vibration.  When  a 
chronograph  is  accessible,  it  is  convenient  to  use  it  in 
obtaining  a  record  of  the  times  of  passage.  Fig.  2 1  is 
from  a  portion  of  such  a  sheet,  upon  which  have  been 
recorded  the  successive  transits  of  a  needle  possessing 
a  period  of  five  seconds.  The  equidistant  notches  are 
the  clock  records.  The  records  of  transit  are  marked 
a,  b,  c,  etc. 

Otherwise  the  period  may  be  determined  with  all 
sufficient  accuracy  by  the  eye  and  ear  method.  In  the 
latter  case  the  observer  counts  the  beats  of  the  chrono- 
meter or  clock  while  watching  the  vibrations  of  the  mag- 
netometer needle,  and  estimates  to  tenths  of  a  second 
the  successive  times  of  transit.  This  estimation  of  tenths 
is  a  matter  requiring  rather  more  practice  in  the  case 


FIG.    21. 

of  time  intervals  than  where  one  has  to  do  with  linear 
measurements  ;  but  it  is  an  art  readily  acquired  by 
practice.  From  the  observed  time  of  vibration  the  true 
period  must  be  computed  by  making  corrections. 

(a)  for  torsion. 

(b)  for  induction. 

(c)  for  temperature. 

(d)  for  the  arc  of  vibration. 

(e)  for  the  rate  of  the  chronometer. 

Under  all  ordinary  circumstances  these  corrections 
are  individually  small,  but  they  are  none  of  them  en- 
tirely negligible. 

The  method  of  correcting  for  torsion  is  similar  to  that 
already  described  in  the  case  of  the  galvanometer,  viz. : 

The  ratio  of  the  torsion  of  the  fibre  to  the  directive 

38 


force  of  the  earth's  magnetic  field  is  determined  by 
twisting  the  head  to  which  the  suspension  fibre  is  at- 
tached through  9oQ,  and  noting  the  deflection  (ti)  of  the 
magnet. 

This  ratio*  is 

£1  11  IQP\ 

7  =  90<nrv 

The  coefficient  of  induction  (//),  by  means  of  which  the 
earth's  inductive  action  in  changing  the  magnetic  con- 
dition of  the  needle  or  bar  is  expressed,  is  the  increase 
in  magnetic  moment  due  to  the  action  of  a  unit  in- 
ducing force. 

Lament's  method  of  determining  p.  consists  in  deter- 
mining the  deflection  produced  by  the  magnet  to  be 
tested  is  placed  vertically,  first,  with  the  north  seeking 
pole  upwards,  and  then  downwards  at  a  constant  dis- 
tance from  a  suspended  needle.  Under  these  condi- 
tions equation  (65)  becomes  respectively 

m~ffF//  =  i  d*  sin  #,  (67) 

and  m+Vfj.  =      ^  ^ 


where  Vis  the  vertical  component  of  the  directive  force 
of  the  earth. 

By  the  combination  of  (67)  and  (68)  we  obtain 

V  ft  _  sin  #'  —  sin  &      tan  j  (#'  —  #)  ,  ~ 

'  ' 


m     ~  sin  &'  +  sin  &  ~  tan  J  (#'  +  0) 

By  making  use  of  the  relation  V  =  H  tan  «,  in  which 
i  is  the  magnetic  inclination,  and  of  the  equation 

—  =  i-  sin  u  which  expresses  the  action  of  the  magnet 
H 

when  used  as  a  deflecting  bar  at  unit  distance,  we  may 
write 

sin  u       tan  ^  (#'  —  d) 
f2  '"  2  tan  i  '  tan  i  (&'  +  #)' 

The  induction  coefficient  is  a  small  quantity.  It  is 
given  by  Figg  and  Whipple,  in  the  case  of  Kew  mag- 
netometer No.  47,  for  example  as  0.0000050.  It  is 
usually  determined  once  for  all  by  the  method  just  de- 
scribed.f 

*  In  this  case  and  in  the  discussion  of  the  subsequent  corrections  the  notation  is  in 
the  main  that  given  by  the  Kew  Observatory  in  the  official  sheets. 
t  See  Stewart  and  Gee  :  Practical  Physics,  11.,  p.  488. 

39 


The  temperature  correction  is  more  important.  The 
bar  to  be  tested  is  placed  in  a  bath,  its  axis  east  and 
west.  Its  deflecting  power  upon  a  magnetometer 
needle  suspended  at  a  suitable  distance  is  noted  at 
various  temperatures  throughout  a  range  of  about 
40 Q  C.  The  influence  of  a  rise  of  temperature  upon 
the  strength  of  magnetization  is  always  to  diminish  it, 
and  the  change  can  be  expressed  by  an  equation  of  the 
form  : 

mt  =  m0  (1  —  a  t  —  I  f).  (71) 

where  mt  is  the  magnetic  moment  at  the  temperature 


i  ooo 


.998 


.996 


.994 


.992 


.990 


TEMP,  COEFF,  or  A  MAGNET 


m 


FIG     22. 

of  observation,  ra0  that  at  op  (7,  and  a  and  b  are  coeffi- 
cients. The  values  of  a  and  b  vary  somewhat  with 
individual  magnets.  The  following  are  average  co- 
efficients: 

a  =  0.000289. 

b  =  0.00500077. 

This  effect  of  temperature  is  not  wholly  temporary. 
If,  for  example,  measurements  be  made  upon  a  mag- 
net, giving  for  rising  temperatures  the  curve  7",  Fig.  22, 
the  succeeding  curve  for  falling  temperatures  (77)  will 


40 


not  coincide  with  it,  but  will  lead  to  a  lower  final  value 
ra0'.  The  amount  of  this  divergence  depends  upon  the 
age,  temper  and  previous  history  of  the  magnet. 

The  corrections  for  tr^e  arc  of  vibration  are  made  by 
means  of  the  well-known  formula 


7   _    T       II 

-*-      •*-  nha    I  J-  r>/»  4  r\f\  ~~~ 


a  a 


In  this  expression  s  is  the  number  of  seconds  gained 
in  a  day  by  the  chronometer  or  clock,  and  a,  a'  are  the 
initial  and  final  values  of  the  semi-arc  of  vibration. 

The  second  operation  of  Gauss  may  be  carried  out  in 
two  ways:  the  first  of  which  is  called  the  method  of 
sines  ;  and  the  second,  the  method  of  tangents.  The 
method  of  tangents  is  the  more  convenient  in  cases  in 
which  the  determinations  are  made  repeatedly  in  a 


FIG.    23. 


single  locality  ;  the  method  of  sines,  on  the  other  hand, 
is  better  adapted  for  portable  instruments.  Of  such 
instruments,  the  Kew  magnetometer  is  the  best  known. 
Its  essential  features  are  a  central  box  or  house,  con- 
taining the  suspended  magnet  C  (Fig.  23),  an  arm  ex- 
tending to  the  north  of  the  axis  of  the  instrument  upon 
which  is  mounted  a  small  reading  telescope  (t)  bearing 
a  short  circular  scale  (s).  By  means  of  this,  observa- 
tions are  made  upon  the  position  of  the  magnetometer 
needle.  A  second  bar  accurately  graduated,  as  to  length, 
extends  to  the  east  and  west  at  right  angles  to  the 
arm  carrying  the  telescope.  Upon  this,  at  fixed  points, 
is  placed,  for  the  purpose  of  the  second  operation,  the 
deflecting  magnet  (Z>).  The  entire  instrument,  includ- 


ing  the  two  arms  just  mentioned  and  the  magnetometer 
box  itself,  are  capable  of  rotation  upon  the  vertical 
axis,  which  axis  corresponds  with  that  of  the  suspended 
magnet.  The  angles,  through  which  this  instrument  is 
turned,  are  measured  upon  a  horizontal  circular  scale 
similar  to  that  with  which  sine  galvanometers  are  pro- 
vided ;  and  the  method  of  making  the  readings  is  the 
same. 

The  expression  made  use  of  in  the  second  operation 
of  Gauss,  see  equation  65,  is  derived  as  follows  : 

Consider  a  magnet,  D  (Fig.  23),  of  strength  of  pole, 


(«*»  -  if 


FIG.   24. 


/,   acting  to   deflect    the    suspended    magnet,    <7,    the 
strength  of  pole  of  which  is  /'. 

The  action  between  pole  S  and  pole  N'  is 


in  which  d  is  the  distance  between  the  centres  of  />  and 
C,  while  I  is  half  the  distance  between  the  poles  of  the 
former. 


The  action  between  JVand  N'  is,  however, 


and  the  total  force  upon  N'  (Fig.  24),  is 

'd         (73) 


The  total  force  upon  S1  is,  in  the  same  way, 


and  the  couple  due  to  the  deflecting  magnet  is 


(d*  _  £)» 
when  Z>  remains  fixed  in  the  east  and  west  direction,  or 

Zff'lld 
(d*  -  ^ 

for  the  case  of  the  Kew  magnetometer. 

When  the  suspended  magnet  has  come  to  rest  at  the 
deflection  #,  we  have 

f:=f'lH,m».  (74) 


Now  the  magnetic  moment  of  the  deflecting  mag 
net  is 


and  when  I  is  small  compared  to  d  (a  condition  which 
should  always  be  fulfilled  in  performing  this  operation) 
we  may  use  the  approximation  (dz  —  i*  =  d*,  and  reduce 
(75)  to  the  same  form  as  (65),  viz.: 

m         d3    . 
Tt  =  "2  sm  * 

The  following  are  the  corrections  to  be  applied  in  the 
computation  of  the  results  of  the  second  operation  : 

(a)  for  temperature  of  the  bar  holding  the  deflecting 
magnet. 

(b)  for  the  distribution  of  magnetism  in  the  same. 

(c)  for  the  temperature  of  the  magnet. 

(d)  for  induction. 

43 


For  the  first  of  these  it  is  sufficient  to  assume  an 
average  coefficient  for  brass  and  to  correct  the  observed 
value  (d')  to  the  proper  value  (dQ)  by  noting  the  tem- 
perature at  which  the  observation  is  made,  and  the  in- 
terval to  the  temperature  at  which  the  scale  upon  the 
bar  is  right. 

The  form  is 

d'  —  dQ  (1  +  0.000018  (f  —  4).  (76) 

The  correction  for  distribution  is  obtained  by  making 
two  sets  of  readings  of  deflection  with  the  magnet  D 
at  very  different  distances  from  C.  The  most  favor- 

dl        3 

able  relation  to  be  that  in  which  -7   =  -;• 

az        4: 

The  correction  for  distribution  (/>)  is  applied  by  means 
of  the  formula 

m         /my  /         p  \ 
~H  ~-:  \H)  \l    "d2)' 

Where  ^  is  the  corrected  and    (  ^i  the    observed 
H  \H) 

value. 

p  is  determined  thus: 

A1  =  i  d?  sin  $! 

A2  =  i  d%  sin  #>.  (78) 

in  which  Al  and  A*  are  the  two  observed  values  of  jr 

at  distances  d^  and  d2. 

From  (77)  it  follows,  however,  that 

/-» \ 

(79) 


or 


.From  equation  80,  which  contains  only  dl7  d2,  ^,and/>, 
the  correction  value  can  be  computed  numerically. 
The  expression  (77)  is  only  an  approximation,  it  is  true, 
but  since  the  value  of  />  is  very  small,  it  is  not  neces- 
sary to  use  the  higher  terms  which  appear  in  a  more 
accurate  formula. 

The  methods  of  obtaining  the  temperature  coefficient 
(q)  and  the  correction  for  induction  (//),  have  already 
been  indicated.  It  should  be  noted,  however,  that  // 
depends  upon  the  position  of  the  magnet  D  in  the  field, 

44 


as  well  as  upon  II.  In  the  tangent  method  of  Gauss  it 
disappears  altogether.  In  the  sine  method,  where  the 
axis  of  D  makes  a  final  angle  ft  with  the  east  and  west 
direction,  the  inductive  effect  on  D  is  p.  II  sin  ft. 

From  this  expression  we  may  obtain  an  approximate 

form  by  use  of  the  equation  sin  $  =          8 . 
The  form  commonly  used  is 

m  —  w0  ( 1  -f-  ~jjf  )  (81) 

in  which  m  is  the  corrected  moment  and  mQ  the  observed 
value. 

Kohlrausch 's  Method  for  H. — Next  in  importance  to 
Gauss'  method  comes  that  of  Kohlrausch,  where  abso- 
lute determinations  of  H  are  desired.  Kohlrausch's 
method,  indeed,  in  those  cases  in  which  the  knowledge 
of  If  is  to  be  used  as  a  factor  in  the  constant  of  the 
galvanometer,  is  much  to  be  preferred  to  any  other, 
because  it  permits  of  the  determination  of  H  in  the 
precise  region  to  which  the  value  is  to  be  applied  ; 
whereas  in  the  case  of  the  Kew  magnetometer  it  is 
oftentimes  necessary  to  make  determinations  which  are 
strictly  applicable  only  to  regions  distant  several  feet 
from  the  precise  locality  from  which  we  desire  to  know 
the  intensity  of  the  field.  The  Kohlrausch  apparatus 
consists  of  the  tangent  galvanometer,  which  may  be, 
and  should  be,  the  galvanometer  for  the  calibration  of 
which  the  value  of  H  is  desired,  and  a  swinging  coil. 
The  coil  is  held  vertically  in  the  magnetic  meridian  by 
means  of  a  bi-filar  suspension.  The  suspension  con- 
sists of  two  wires  which  serve  to  carry  current  into  and 
out  of  the  suspended  coil,  and  at  the  same  time  give  it 
the  necessary  directive  force.  This  coil  should  be  sus- 
pended as  nearly  as  possible  in  the  region  containing 
the  needle  of  the  galvanometer  which  we  wish  to  cali- 
brate. In  the  case  of  tangent  galvanometers  of  the 
Helmholtz  pattern,  it  is  oftentimes  entirely  practicable 
to  mount  the  swinging  coil  midway  between  the  fixed 
coils  of  the  galvanometer,  so  that  the  needle  will  be 
within  the  plane  of  the  former  and  in  its  axis.  Such 
an  arrangement  exists  in  the  large  tangent  galvano- 
meter described  in  the  first  lecture.  The  method  of 
procedure  is  as  follows  :  A  current  i  is  sent  through  the 
coils  of  a  galvanometer  of  known  dimensions,  and  in 
the  case  in  question  this  should  be  the  galvanometer  to 
be  calibrated.  We  have  then 

i  =  -^  tan  #.  (82) 

45 


The  same  current  is  then  sent  through  the  suspended 
coil.  If  the  galvanometer  coils  or  the  suspended  coils 
are  of  appreciable  resistance,  as  compared  with  the  re- 
mainder of  the  circuit,  it  is  necessary  to  have  elsewhere 
resistances  which  can  be  inserted  and  removed,  the 
values  of  which  have  been  previously  adjusted  so  as  to 
correspond  precisely  to  the  resistance  of  the  galvano- 
meter and  of  the  swinging  coil  respectively.  A  con- 
venient arrangement  for  doing  this  is  represented  in 
Fig.  25,  in  which  fig  and  Ka  are  resistances,  non-in- 
ductively  wound,  which  are  to  be  substituted  in  turn 
for  the  galvanometer  and  for  the  swinging  coil  by 
means  of  the  switch  S.  When  the  current  i  is  sent 
through  the  swinging  coil  it  will  deflect  the  latter.  This 
deflection  is  to  be  read  by  means  of  a  suitably  adjusted 
telescope  and  scale.  We  have  then 


^  =  : 


, 
tan  a 


(83) 


FIG.    25. 

in  which  Mb  is  the  moment  of  the  bi-filar  suspension,  A 
is  the  effective  area  of  the  suspended  coil,  the  dimen- 
sions of  which  must  have  been  determined  by  previous 
measurement,  and  «  is  the  deflection  of  the  coil  from 
the  magnetic  meridian.  By  combining  equations  82 
and  83,  we  obtain  the  following  expression  for  IIZ  in 
terms  of  the  constants  of  the  galvanometer  and  of  the 
swinging  coil,  the  moment  of  the  latter  and  the  ratio  of 
the  tangents  of  the  deflections,  viz.: 


In  the  hands  of  Kohlrausch  this  method  has  afforded 
some  of  the  most  precise  determinations  of  the  abso- 

46 


lute  value  of  the  horizontal  component  of  the  earth's 
magnetic  field  which  have  ever  been  made.  He  availed 
himself  of  it,  for  example,  in  connection  with  his  re-de- 
terminations of  the  electro- chemical  equivalents  of 
silver  and  copper.  Such  operations  demand  the  most 
precise  knowledge  of  all  the  factors  which  enter  into 
the  operation. 

The  most  troublesome  features  of  this  method  are 
those  which  dealwith  the  temperature  changes  of  J/h, 
the  secular  changes  of  the  same  and  variation  in  the 
resistance  of  the  suspension  wires.  These  wires  to  give 
sufficient  delicacy  to  the  method  must  be  of  small  size, 
and  they  must  consequently  be  Subjected  to  high 
current  densities. 

In  the  case  of  the  large  tangent  galvanometer  at 
Cornell  University,  reference  to  which  was  made  in  the 
first  lecture,  an  arrangement  was  perfected  for  the  de- 
termination of  H  by  a  modification  of  the  method  of 
Kohlrausch.  A  large  swinging  coil,  the  diameter  of 
which  was  one  meter,  was  suspended  by  means  of  a 
phosphor  bronze  wire  two  meters  long,  the  upper  end 
of  which  was  attached  to  a  torsion  head  carrying  a  circle 
and  vernier.  The  method  consists  in  bringing  the 
swinging  coil  back  to  its  zero  position  by  twisting  the 
suspension  wire.  The  position  of  the  coil  was  read  by 
means  of  a  telescope  and  scale  at  a  distance  of  three 
meters  to  the  south  of  the  instrument,  for  which  pur- 
pose the  torsion  head  could  be  given  a  slow  motion  of 
rotation  by  means  of  a  tangent  screw  operated  by  the 
observer  at  the  reading  telescope.  The  suspension 
wire  served  also  to  introduce  current  to  the  suspended 
coil,  which  consisted  of  100  turns  of  No.  18  copper 
wire.  The  other  terminal  of  the  coil  consisted  of  a 
wire  situated  in  the  axis  of  rotation,  the  end  of  which 
was  dipped  in  a  mercury  cup  at  the  base  of  the  instru- 
ment. This  form  of  suspension  gave  greater  delicacy 
than  could  be  obtained  by  means  of  any  bi-filar  suspen- 
sion, which  would  be  capable  of  carrying  the  currents 
which  it  was  necessary  to  introduce  into  the  coil.  The 
method  possessed  also  the  advantages  common  to  what 
are  known  as  zero  methods.  The  equations,  by  means  of 
which  H  is  determined  with  this  instrument,  differ  from 
those  which  apply  to  the  Kohlrausch  method  only  in 
two  particulars.  In  the  first  place,  the  force  of  torsion 
used  in  returning  the  wire  to  its  original  position  is  pro- 
portional to  the  angle  through  which  the  suspension  wire 
is  twisted.  In  the  second  place,  we  have  to  substitute 
for  the  moment  of  the  bi-filar  suspension  the  moment  of 
torsion  of  the  wire.  Making  these  changes  in  equation 
83,  we  have  the  following  : 

47 


t 

^  =:  jf-jfO,  (85) 

IP  __  GM*  .  _JL_  (86) 

A        tan  # 

The  moment  of  torsion  is  determined  by  substituting 
for  the  swinging  coil,  which  is  so  adjusted  as  to  be 
readily  unmounted  and  removed  from  its  position,  a 
cylindrical  brass  weight  the  moment  of  inertia  of  which 
can  be  determined  directly  from  its  mass  and  from  its 
dimensions.  This  cylinder,  in  the  instrument  under 
consideration,  weighs  4954.22  grammes,  and  its  diameter 
is  7.8747  cm.  The'weight  is  hung  in  place  of  the  coil 
and  its  period  of  oscillation  is  accurately  determined 
by  the  aid  of  the  chronograph.  The  expression  for  the 
moment  of  torsion  takes  the  usual  form  of  the  equation 
for  the  torsion  pendulum,  viz.: 

"  •;         ;•  (87) 

in  which,  when  we  know,  the  moment  of  inertia  K  and 
the  period  of  oscillation  T,  we  have  all  the  factors 
necessary  to  the  computation  of  H  in  absolute  measure. 
Substituting  in  equation  (86)  the  above  value  of  the 
moment  of  torsion,  we  have 


or 


(88) 


The  quantity  C  in  equation  (88)  is  a  constant  such 
that  : 


When  this  .method  was  first  put  into  operation  in 
1886,  two  serious  sources  of  error,  one  of  which  was  en- 
tirely unexpected,  arose.  The  first  of  these  was  an 
error  due  to  the  influence  of  temperature  upon  the 
moment  of  torsion  of  the  suspension  wire.  This  error 
has  its  basis  in  a  property  of  matter  which  is  perfectly 
well  known.  It  is  not  a  difficult  matter  to  determine  once 
for  all  the  temperature  coefficient  of  torsion  for  the 
material  used,  and  to  apply  the  correction.  The  diffi- 
culty in  maintaining  a  vertical  wire,  two  meters  long, 
at  anything  approximating  a  constant  temperature 
throughout  its  entire  length,  however,  was  found  to  be 
unsurmountable  under  the  conditions  which  existed  in 
the  observatory  where  the  galvanometer  was  situated, 

48 


and  no  satisfactory  correction  for  the  temperature  of 
the  wire  was  reached  until  after  many  expedients  had 
been  tried  and  abandoned ;  the  following  method  of 
ascertaining  the  average  temperature  of  the  wire  at  the 
precise  time  when  each  observation  was  made  came  to 
be  adopted. 

This  method  of  integrating  the  temperature  for 
the  entire  length  of  the  wire  consisted  in  placing  a 
No.  40  copper  wire  parallel  to  the  suspension  and  as 
close  to  the  same  as  could  be  without  actual  contact. 
This  copper  wire  was  drawn  back  and  forth  several 
times.  It  was  placed  in  series  with  a  compensated  re- 
sistance, the  value  of  which  was  approximately  the 


SECONDS 


8.704  8.708  8.712  8-7l« 

FIG.    26. 

same  as  its  own  and  a  suitable  current  sent  through  the 
circuit  containing  the  two.  A  sensitive  galvanometer 
was  mounted  in  another  part  of  the  observatory,  by 
means  of  which  the  flow  of  potential  through  the  com- 
pensated resistance  and  through  the  temperature  wire 
just  described  could  be  compared.  The  ratio  of  these 
deflections  gave  the  average  temperature  of  the  wire 
with  a  high  degree  of  accuracy. 

Before  mounting  the  fine  copper  for  this  purpose,  its 
temperature  coefficient  had  been  determined,  and  the 
ratio  of  the  deflections  when  the  galvanometer  was 


49 


shunted  across  its  terminal,  to  that  obtained  when  the 
galvanometer  was  shunted  across  the  terminals  of  the 
compensated  resistance,  had  been  ascertained  for  a 
sufficient  range  of  temperatures.  By  the  use  of  this 
simple  device,  the  difficulties  arising  from  difference  of 
temperatures  in  the  suspension  wire  were  eliminated. 

The  other  source  of  error  was  of  a  more  serious 
character.  It  was  found  that  the  torsional  elasticity  of 
the  suspension  wire  varied  continually  with  age.  The 
change,  which  was  very  marked,  indeed,  at  first,  dimin- 
ished slowly  as  time  passed  ;  but  it  never  became  a 
negligible  quantity.  The  time  curve  of  this  wire  has 
been  taken  with  great  care,  and  it  now  covers  an  inter- 
val of  nearly  ten  years.  By  means  of  this  curve  the  mo- 
ment of  torsion  of  the  wire  for  any  desired  date  can  be 
ascertained  with  a  sufficient  degree  of  accuracy;  but  with- 


FIG.  27. 

out  this  correction  the  values  for  H  determined  by  this 
method  would  be  seriously  at  fault.  The  character  of 
the  first  of  these  two  variations,  that  due  to  tempera- 
ture and  is  shown  graphically  in  Fig.  26,  which  gives 
the  relation  of  the  rate  of  vibration  of  the  wire  when 
attached  to  the  calibration  weight  already  described. 
This  curve  was  made  on  February  26,  1887,  and  is  from 
measurements  by  Professor  H.  J.  Ryan.  Determina- 
tions at  later  dates  would  afford  data  showing  a  slower 
period.  The  rate  at  10°,  on  October  28,  1889;  for  ex- 
ample, according  to  measurements  by  Mr.  N.  H. 
Genung  was  8.689  +  seconds. 

To  control  these  factors,  upon  which  accuracy  de- 
pends, is  a  matter  of  considerable  difficulty,  and  the 
method  of  Kohlrausch  for  the  determination  of  H  is 
rendered  a  laborious  one  because  of  them. 

50 


The  Determination  of  H  by  Means  of  the  Copper  Volta- 
meter.— For  all  ordinary  operations  with  the  tangent 
galvanometer,  a  sufficiently  accurate  determination  of 
H  can  be  obtained  by  a  method  which  is  much  more 
convenient  than  those  of  Gauss  or  Kohlrausch.  This 
method  consists  in  sending  through  the  galvanometer 
a  current,  the  intensity  of  which  is  measured  by  means 
of  copper  voltameters  placed  in  the  circuit,  and  noting 
the  deflection  produced.  The  requisites  are  a  steady 
source  of  current,  such  as  a  storage  battery  of  consider- 
able capacity,  a  fine  balance,  a  fairly  accurate  time- 
piece and  a  copper  voltameter  of  proper  construction. 
The  form  of  voltameter  which  has  shown  itself  best 
adapted  to  accurate  work  is  one  which  is  at  the  same 
time  the  most  easily  constructed.  I  refer  to  the  spiral 
coil  voltameter  of  Professor  Ryan.*  This  consists  of 
two  suitable  jars  containing  a  slightly  acidulated  solu- 
tion of  the  sulphate  of  copper,  two  coils  of  pure  copper 
wire  about  five  centimeters  in  diameter  which  are  to 
form  the  losing  electrodes,  two  coils  of  smaller  diameter 


FIG.  28. 

constructed  from  the  same  wire,  which  are  to  constitute 
the  gaining  electrodes,  and  any  simple  device  for  hold- 
ing these  coils  pair-wise  in  the  cells  with  a  common 
vertical  axis  corresponding  to  the  axis  of  the  jar  (see 
Fig.  27).  To  construct  these  coils,  it  is  only  necessary 
to  take  a  few  feet  of  copper,  size  No.  10  or  No.  12,  to 
strip  the  same  of  its  isolation,  to  clean  the  wire  thor- 
oughly by  clamping  one  end  of  it  and  drawing  sand 
paper,  grasped  in  the  hand,  briskly,  over  its  entire  length 
several  times.  The  wire  is  then  wound  upon  cylinders 
of  suitable  diameter  so  as  to  form  two  large  and  small 
coils  such  as  have  been  already  described.  When  com- 
pleted, these  coils  will  have  a  length  along  the  axis  of 
about  three  inches,  and  their  diameters  will  be  respec- 
tively for  the  losing  coils,  five,  and  for  the  gaining  coils, 
two  centimeters. 

The  smaller  coils  are  carefully  weighed  upon  a  bal- 
ance of  high  precision,  are  then   mounted  within  the 


large  coils  in  the  two  jars,  Fig.  28,  and  concentric  with 
the  same  ;  the  jars  are  filled  with  the  electrolytic  solu- 
tion, electric  connections  are  completed,  and  the  time 
of  making  circuit  is  carefully  noted.  In  the  course  of 
a  half  hour  or  thereabouts,  during  which  the  current  is 
flowing  through  the  cells  and  through  the  galvano- 
meter, a  number  of  readings  of  the  deflection  are  taken. 
Th  e  current  is  then  broken  at  a  time  accurately  noted, 
and  the  inner  or  gaining  coils  are  removed  from  the 
voltameter.  They  are  rinsed  with  water  and  then  with 
alcohol,  after  which  they  are  dried  without  friction  with 
filter  paper,  or  by  holding  them  at  a  safe  distance  over 
the  flame  of  a  bunsen  burner. 

If  the  operation  has  been  a  successful  one,  the  sur- 
face will  possess  a  uniform  and  beautifully  tinted  sur- 
face characteristic  of  freshly  deposited  electrolytic 
copper.  Any  marked  granulation  of  the  surface  would 
indicate  too  great  a  current  density,  and  would  subject 
the  results  of  the  measurement  to  suspicion.  Under  such 
circumstances  the  calibration  should  be  repeated.  The 
amount  of  copper  deposited  upon  two  plates,  as  shown 
by  the  comparison  of  the  weighings  before  and  after 
should  agree  to  within  two-tenths  of  one  per  cent.  Pro 
perly  carried  out,  therefore,  this  method  will  give  the 
value  of  H  to  a  like  degree  of  precision. 


LECTURE  V. 
GALVANOMETERS  WITH  ARTIFICIAL  FIELDS. 

i.  Instruments  with  Strong  Fields . — The  magnetic  field 
of  the  galvanometer  is  frequently  strengthened  artifici- 
ally for  one  or  more  of  the  following  reasons: 

(a)  To  increase  the  constant  — ,  thereby  securing  an 

instrument  suitable  to  the  measurement  of  heavy  cur- 
rents. 

(b)  To  diminish  the  period  of  vibration  of  the  needle. 

(c)  To  obtain  immunity  from  magnetic  disturbances, 
such  as  the  daily  fluctuations  which  take  place  in  the 
value  of  JET,  and  the  accidental  variations  brought  about 
by  the  proximity  of  masses  of  iron  or  by  the  inductive 
influence   of  the   dynamo    motors,   and   of    line   wires 
carrying  current. 

The  most  important  instruments  of  the  kind  under 
consideration  are  the  galvanometers  of  the  D' Arson val 
type.  In  these  well-known  galvanometers  a  strong  field 
is  obtained  by  means  of  a  nearly  closed  magnetic  circuit. 
Within  the  air  space  of  this  magnetic  circuit  is  placed  a 
coil  of  wire,  through  which  the  current  to  be  measured 
is  allowed  to  pass.  This  coil  has  freedom  of  rotation 
upon  an  axis  at  right  angles  to  the  lines  of  force,  and 
also  at  right  angles  to  the  axis  the  coil  itself.  In  order 
to  hold  such  a  coil  in  place  in  the  very  strong  fields 
which  are  made  use  of  in  these  instruments  the  suspen- 
sion is  by  means  of  wires  vertically  fastened  above  and 
below. 

The  original  type  described  by  Deprez  and  D'Arson- 
val,*  is  shown  in  Fig.  29.  A  coil  thus  held  between 
tense  suspension  wires  vibrates  rapidly,  and  a  short 
period  of  oscillation  is,  therefore,  one  of  the  character- 
istic features  of  such  instruments. 

The  D'Arsonval  galvanometer  has  been  subjected  to 
a  great  variety  of  modifications.  We  have,  for  example, 
the  moving  coil  without  an  iron  core,  a  moving  coil 

*  Deprez  and  D'Arsonval:  Comptes  Rendus  94,  p.  1347,  1882. 

53 


with  an  iron  core,  a  moving  coil  the  interior  of  which 
is  rilled  with  a  stationary  piece  of  soft  iron.  The 
field  in  which  the  coil  swings  is  sometimes  that  of  a 
permanent  magnet  of  the  horseshoe  type,  sometimes 
that  of  an  electromagnet,  and  sometimes  that  of  a  sole- 
noid without  iron.  The  air  gap  also  is  of  various  sizes, 
from  the  very  large  air  gap  of  the  Thomson  graded 
galvanometer  to  the  exceedingly  small  one  employed 
in  instruments  of  the  Breguet  form.  The  advantages 


FIG.   29. 

of  all  these  galvanometers  may  be  summed  up  in  the 
statement  that  they  are  exceedingly  quick  of  action 
and  remarkably  free  from  outside  influences.  As  to  the 
permanency  of  their  indications,  it  is  evident  that  any 
instrument,  the  constant  of  which  depends  upon  the 
maintenance  of  an  unchanged  field  due  to  permanent 
magnets,  must  be  subject  to  a  certain  amount  of  secular 
change.  Whether  the  time  change  in  the  field  of  such 
instruments  can  be  reduced  to  an  inappreciable  quan- 


54 


tity  is  a  subject  about  which  there  has  been  consider- 
able discussion.  Dr.  Koepsel,*  for  example,  in  a  paper 
read  before  the  Electro-technical  Congress  at  Frank- 
fort, in  1891,  took  the  ground  that  the  use  of  steel  mag- 
nets in  instruments  for  the  measurement  of  electric 
current  should  be  altogether  abandoned  on  account  of 
the  lacK  of  permanence.  This  view  has  been  combatted, 
however,  on  the  part  of  those  who  have  had  much  ex- 
perience in  making  instruments  in  which  permanent 
magnets  are  used.  The  permanence  of  such  magnets, 
undoubtedly,  depends  in  part  on  the  size  of  the  air  gap, 
and  increases  as  the  latter  is  reduced. 

Instruments  such  as  the  Thomson  graded  galvanome- 
ters, on  the  one  hand,  in  which  the  magnetic  circuit  is 
nearly  half  through  the  air,  exhibit  much  more  rapid  de- 
cadence than  instruments  of  the  Deprez  type  in  which 
the  air  gap  is  reduced  to  a  minimum.  A  graded  galva- 
nometer with  a  home-made  magnet  which  had  been  con- 
structed to  take  the  place  of  the  original  magnet  be- 
longing to  the  instrument  showed,  for  example,  a  change 
of  constant  in  one  year  from  7.00  to  6.6 1.  This  marked 
falling  off  in  the  strength  of  the  field  may,  with  justice, 
be  ascribed  in  part  to  the  inadequate  treatment  of  the 
permanent  magnet  in  preparing  it  for  use  in  such  an 
instrument.  The  original  magnet  belonging  to  a  simi- 
lar galvanometer  showed,  however,  a  scarcely  better  re- 
cord. The  constant  fell  off  in  this  second  case  from 
4.52  to  4.44  in  one  year.  An  ammeter  of  the  D' Arson  - 
val- Deprez  type  showed  somewhat  greater  permanence. 
A  current,  which,  on  the  3oth  of  November,  1892,  pro- 
duced a  deflection  of 

36.2  scale  divisions  to  the  right, 

35.8  scale  divisions  to  the  left, 

was  found  in  October,  1893,  to  produce,  respectively 
35.1  scale  divisions  to  the  right, 

34.9  scale  divisions  to  the  left. 

As  originally  constructed,  the  calibration  curves  of 
the  D'Arsonval  galvanometer  were  by  no  means 
straight.  Figure  30  shows  the  curve  for  right  and  left 
deflections  in  the  case  of  the  ammeter  just  referred  to. 
It  is  obvious,  however,  that  the  law  of  deflections  in  all 
such  instruments  is. under  control  by  modifying  the 
shape  and  disposition  of  the  pole  pieces.  Professors 
Ayrton  and  Perry  have  shown  that  by  this  device  the 
curve  can  be  readily  straightened. 

An  interesting  example  of  the  application  of  the  prin- 
ciples upon  which  galvanometers  with  strong  fields  de- 

*  Koepsel :  Verhandlungen  des  internationalen  Elektrotechniker-Consrresses  zu 
Frankfurt,  1892,  Zweite  Halfte,  p.  3. 

55 


pend  is  found  in  the  Moler  curve  writing  voltmeter.* 
This  is  essentially  a  galvanometer  of  the  D'Arsonval 
type  in  which  a  needle  of  soft  iron  is  mounted  in  the 
strong  field  between  the  poles  of  a  powerful  permanent 
magnet. 

The  needle  carries  a  short  aluminium  pointer  which 
records  its  oscillations  upon  the  smoked  drum,  Fig.  3 1 . 
The  vibrations  of  the  needle  of  this  instrument  are  so 
rapid  that  by  means  of  it  one  can  follow  the  fluctuations 
of  current  during  a  single  revolution  of  a  dynamo  or 
motor.  The  amplitude  of  vibration  is  necessarily  quite 
small,  but  the  indications  of  the  instrument  are  never- 
theless exact,  and  when  measured  and  duly  magnified 
they  are  found  to  correspond  excellently  with  results 
obtained  by  other  methods. 

It  is  possible,  by  means  of  an  instrument  of  this  kind, 
to  make  interesting  studies  of  a  great  variety  of  pheno- 


FIG.    30. 

mena  in  which  rapid  fluctuations  of  current  occur.  The 
instrument  was  originally  devised  for  the  purpose  of 
exploring  the  field  of  dynamos  and  motors,  a  purpose 
to  which  it  is  well  adapted.  It  can,  however,  be  used  in 
many  other  ways. 

Fig.  32  shows  the  tracing  obtained  by  means  of  this 
instrument  in  the  study  of  the  performance  of  arc 
lamps.  The  voltmeter  was  placed  across  the  terminals 
of  a  direct  current  lamp  the  construction  of  which  is 
such  that  the  carbons  are  held  apart  by  a  spring  until 
they  are  brought  together  by  the  action  of  the  current 
through  the  shunt  coil,  after  which  they  are  separated 
again  and  the  arc  is  formed.  The  point  marked  w  in  the 
figure  is  that  at  which  the  circuit  was  closed.  The  ver- 
tical distance  between  the  upper  and  lower  tracings  at 

*  American  Institute  of  Electrical  Engineers,  vol.  9,  p.  223. 


NI  v 


b  is  50  volts  which  is  the  normal  potential  difference  of 
the  lamp.  It  will  be  seen  that  immediately  after  clos- 
ing- the  circuit  atw  there  is  a  rise  of  potential  to  a  much 
higher  value  during  the  interval  before  the  shunt  coil 
comes  into  operation  ;  furthermore,  that  the  carbons  do 
not  come  into  contact  for  a  considerable  time,  viz.,  that 
which  elapses  between  the  point  marked  w  and  that 


f*       D 


FIG.  31. 

marked  x.  There  is  also  a  further  interval  of  time  from 
x  to  y,  during  which  the  contact  between  the  carbons, 
which  had  been  poor  at  first,  improved  as  the  tips  grew 
hot.  This  appears  from  the  fact  that  the  potential  dif- 
ference continues  to  fall  off  until  y  is  reached,  at  which 
time  it  is  virtually  reduced  to  zero.  Finally,  the  adjust- 
ing mechanism  of  the  lamp  begins  to  act,  and  the  car- 
bons are  drawn  apart  to  their  normal  condition. 


FIG.  32. 

2.  Instruments  with  Weak  Fields. — A  more  important 
modification  of  the  field  of  the  galvanometer  than  that 
which  we  have  just  considered,  is  in  the  direction  of 
reducing  its  intensity.  The  object  sought  in  such  cases 
is  increase  of  sensitiveness.  The  limit  to  which  such 
increase  can  be  carried  is  reached  only  when  the  natural 
or  artificial  changes  of  the  field  thus  produced  become 

57 


so  great  as  to  cause  troublesome  drifting  of  the  gal- 
vanometer needle.  A  magnetized  needle  in  the  weak- 
ened field  shows  motions  corresponding  to  those  of  the 
declination  needle,  but  the  amplitude  of  fluctuation  is 
increased.  The  sensitized  galvanometer,  therefore, 
drifts  more  and  more  as  its  sensitiveness  increases. 

The  following  simple  demonstration  of  the  relation 
between  sensitiveness  and  the  fluctuation  of  the  direc- 
tion of  the  resultant  force  of  the  weakened  field  is  due 
to  Mr.  F.  J.  Rogers.*  In  Fig.  33,  let  o  E  represent  the 
direction  and  the  directive  force  of  the  earth's  magnetic 
field  which  has  been  partly  counteracted  by  the  intro- 
duction into  the  neighborhood  of  a  controlling  magnet 
which  produces  a  field  represented  by  the  line  o  M. 
The  resultant  field  is  o  R.  If  now  from  any  cause  the 


original  field  be  subjected  to  a  change  of  direction  and 
intensity  that  it  must  be  represented  by  o  E',  which 
makes  an  angle  a  with  o  E,  the  new  resultant  formed 
by  the  combination  of  o  M  with  o  E'  will  take  a  new 
direction  o  R',  which  makes  a  much  larger  angle  a!  with 
the  first  resultant  o  R.  To  the  same  writer  is  due  a 
very  interesting  experimental  study  of  the  behavior  of 
sensitive  galvanometers. 

For  this  purpose  two  galvanometers  were  taken,  one 
of  which  was  highly  sensitized  by  the  action  of  a  con- 
trolling magnet,  while  the  other  possessed  a  field  due  to 
the  uncounteracted  intensity  of  the  earth's  magnetism 
at  the  point  at  which  the  instrument  was  set  up.  The 

*  F.  J.  Rogers,  The  Crank^  vol.  6,  1892,  p.  270. 

58 


first  of  these  was  studied  on  three  successive  days  for 
the  purpose  of  observing  the  range  of  fluctuation  of 
the  zero  point  under  the  ordinary  changes  in  the  direc- 
tion of  the  earth's  lines. 

The  course  followed  by  the  needle  is  shown  in  Fig. 
34.  It  was  the  same  in  all  essential  particulars  during 
the  three  days  in  question,  reaching  a  maximum  of 
elongation  from  its  mean  position  daily  just  before 
2  p.  M.  These  curves  correspond  very  closely  with 
those  which  would  be  obtained  by  the  observation  of 
the  movement  of  a  declination  needle  at  times  when 
there  was  no  marked  magnetic  storm.  The  amplitudes 
of  oscillation,  however,  are  very  much  greater  owing 
to  the  weakness  of  the  field.  On  each  of  these  days 
the  other  galvanometer,  the  needle  of  which  was  sus- 
pended in  the  earth's  field,  went  through  the  same 
range  with  about  one-twentieth  of  the  amplitude  shown 
in  Fig.  34. 


74  s  d. 


66  s,  d. 


X 


58  s. 


6    P.M. 


FIG.   34. 

To  further  establish  the  relationship  between  the 
movements  of  a  sensitive  galvanometer  and  those  of 
the  declination  needle,  in  other  words,  to  ascertain 
whether  the  movements,  familiar  to  all  who  use  sensi- 
tive instruments  of  the  class  in  question,  are  due  to 
local  disturbances,  or  to  those  widespread  magnetic 
fluctuations  which  cause  magnetometer  needles  over 
the  entire  continent  to  move  together,  observations 
were  made  with  the  two  instruments  already  described 
upon  a  day  when  there  was  marked  magnetic  disturb- 
ance. The  result  is  shown  in  Fig.  35. 

In  this  diagram  the  amplitude  of  fluctuation  of  the 
needle,  which  was  suspended  in  the  earth's  field,  was 
multiplied  by  a  constant  factor  such  as  to  bring  it  to 
the  same  scale  as  that  of  the  more  sensitive  galvano- 
meter. To  make  sure  that  the  fluctuations  which  af- 
fected the  two  galvanometers  simultaneously  through 


59 


the  day,  although  they  were  mounted  in  different  rooms 
of  the  laboratory,  were  not  due  to  local  causes,  but 
were  the  result  of  changes  in  the  magnetic  field  of  the 
earth  itself,  records  were  obtained  for  the  day  in  ques- 
tion from  the  magnetic  observatory  at  Washington. 
These  were  multiplied  by  the  proper  factor  to  bring 
them  to  the  exaggerated  scale  applicable  to  the  sensi- 
tized galvanometer,  and  were  plotted  upon  the  same 
sheet.  In  Fig.  35,  to  which  reference  has  just  been 
made,  the  unbroken  line  represents  the  changes  in  the 
position  of  the  declination  needle  as  recorded  at  Wash- 
ington on  the  day  in  question.  The  two  broken 
curves  are  those  obtained  by  making  readings  with  the 
two  galvanometers  in  the  laboratory  at  Ithaca.  It  will 
be  seen  that  the  three  curves  agree  in  a  remarkable 
manner  throughout. 

The  consideration  of  these  results  makes  it  obvious 
that  in  order  to  use  a  sensitive  galvanometer  in  opera- 
tions of  precision,  two  things  are  necessary: 


FIG.    35. 

1.  A  knowledge  of  Hm  the  locality  where  the  instru- 
ment is  in  use  at  the  time  when  calibration  of  the  latter 
is  made. 

2.  Some  method  of  following  the  fluctuations  of  H 
from  moment  to  moment. 

A  discussion  of  the  means  of  meeting  this  second 
requirement  will  be  given  in  a  subsequent  lecture. 

The  methods  of  determining  H,  described  in  the 
fourth  lecture  are  very  laborious,  and  it  is  desirable, 
therefore,  to  substitute  for  them  some  means  for  com- 
paring the  strength  of  unknown  fields  with  that  of 
known  fields  previously  determined  for  this  purpose  the 
method  of  Wilhelm  Weber  is  most  convenient.  It  is 
described  below. 


Weber's  Method  for  the  Determination  of  H. — The  pro- 
cedure consists  in  turning  a  coil  of  known  dimensions 
(the  earth  inductor)  suddenly  through  180°,  and  noting 


60 


the  throw  of  the  slow  moving  needle  of  a  ballistic  gal- 
ganometer  placed  in  circuit  with  this  coil.  Fig.  36  shows 
the  instrument  in  its  simplest  form,  while  Fig.  37  gives 
the  arrangement  of  the  electrical  circuit. 

The  earth  inductor  is  a  coil  of  considerable  area,  and 
consisting  of  many  turns  of  copper  wire.  The  dimen- 
sions of  the  coil,  the  number  of  turns  and  the  resist- 
ance of  the  instrument  will  depend  upon  the  character 
of  the  galvanometer  with  which  it  is  to  be  used. 

The  coil  is  mounted  in  a  strong  wooden  frame  with 


FIG.  36. 


freedom  to  revolve  upon  a  vertical  axis.  Stops  are  so 
placed  as  to  limit  this  motion  to  exactly  180°.  The 
frame  itself  is  free  to  turn  upon  a  horizontal  axis,  with 
stops  to  facilitate  the  adjustment  of  it  in  a  horizontal  or 
in  a  vertical  plane. 

The  base  of  the  earth  inductor  should  be  provided 
with  a  good  level,  and  with  levelling  screws.  In  the 
operations  to  be  considered  here  it  is  desired  to  com- 
pare the  value  of  S in  a  locality  where  that  quantity  is 
known,  as,  for  example,  in  the  magnetic  observatory, 

61 


with  the  value  of  H  where  the  galvanometer  is  mounted. 
For  this  purpose  the  earth  inductor  is  carefully  levelled 
in  the  former  locality  with  its  coil  vertical  and  against 
the  stops,  the  axis  of  the  coil  being  in  the  magnetic 
meridian.  It  is  connected  with  a  line  leading  to  the 
galvanometer.  The  earth  inductor  is  in  series  with  the 
latter  and  with  a  resistance  R.  These  three  portions  of 
the  circuit,  viz.,  the  coils  of  the  galvanometer  of  the 
earth  inductor  and  of  the  resistance  R,  should  include 
nearly  all  the  resistance,  that  of  the  line  being  negligible. 

The  circuit  having  been  completed,  the  observer 
watches  the  galvanometer  while  an  assistant  swings  the 
earth  inductor  through  180°.  A  series  of  readings  are 
thus  made,  using  the  galvanometer  ballistically. 

It  is  necessary  to  accuracy,  that  the  period  of  vibra- 
tion of  the  instrument  be  much  longer  than  the  inter- 
val of  time  occupied  by  the  semi-revolution  of  the 
inductor.  Otherwise  one  of  the  conditions  indicated  in 
Lecture  III  ,  viz.,  that  the  entire  quantity  of  electricity 
due  to  the  motion  of  the  coil  should  traverse  the  coils 
of  the  galvanometer  before  the  needle  had  moved 
through  a  considerable  angle,  will  not  be  fulfilled. 


FIG.  37. 

After  the  completion  of  a  series  of  readings  the  earth 
inductor  is  removed  to  the  spot  in  which  the  intensity 
of  the  field  is  to  be  compared  with  that  of  the  locality 
which  that  instrument  had  occupied.  After  proper  ad- 
justment of  the  coil  in  its  new  place,  a  new  series  of 
semi-revolutions  is  made  and  the  corresponding  deflec- 
tions are  noted. 

It  is  by  the  comparison  of  these  deflections  that  the 
two  magnetic  fields  are  brought  into  relative  measure- 
ment. The  changes  of  the  field  in  the  second  locality 
can,  moreover,  be  followed  by  repetition  from  time  to 
time  of  the  second  series  of  determinations. 

The  expression  for  quantity  of  electricity  generated 
by  the  motion  of  the  earth  inductor  is 


Q  =        ^      %  sin  J  #  =  2  a  tfg  jsin  \  &  (90) 

an  equation,  the  development  of  which  has  been  given 

62 


in  the  lecture  on  the  ballistic  galvanometer  The 
quantity  §,  however,  depends  upon  the  area  A  of  the 
earth  inductor  coil,  and  the  resistance  R  of  the  entire 
circuit  as  well  as  upon  the  strength  of  the  field  within 
which  the  rotation  of  the  coil  occurs.  This  relationship 
is  expressed  by  the  formula 


combining  equations  90  and  91  we  may  write 


This  equation  may  be  used  in  several  ways  : 

1.  Given  #e,  A,  R,  G  and  T, 

HK,  the  strength  of  the  field  in  which  the  galvanometer 
needle  swings  may  be  determined. 

2.  Given  the   constant  of    the  galvanometer   in   the 
field  Hg  also  A  and  R,  any  field  ffe  in  which  the  earth 
inductor  is  turned  may  be  computed  in  absolute  meas- 
ure. 

3.  The  operations  just  described  yield  two  equations 
of  the  type  90,  viz.: 


sin^,  (94) 

By  combining  these  we  are  able  to  eliminate  all  the 
constants,  and  to  obtain  a  ratio  between  the  fields  to  be 
explored,  viz.: 

ff.  _  sin  j  #e  (95) 

Hx  ~  sin  £  #x 

The  method  of  the  earth  inductor  is  especially  useful 
when  a  very  strong  field,  such  as>  that  which  exists  in 
the  air  gap  of  an  electromagnet,  is  to  be  compared 
with  He. 

For  such  measurements  a  coil  should  be  constructed, 
the  cross-section  of  which  is  not  too  great  to  admit  it 
entirely  within  the  field  to  be  measured. 

The  total  area  of  this  coil  is  to  be  determined  as 
accurately  as  possible  at  the  time  of  winding. 

It  is  generally  better  to  pull  this  coil  out  of  the  field 
than  to  attempt  to  give  it  a  motion  of  rotation.  Some- 
times this  can  be  most  conveniently  accomplished  by 
the  aid  of  gravitation,  the  coil  being  attached  to  a 
weight,  as  in  Fig.  38,  and  released  by  breaking  the  cir- 
cuit which  animates  the  small  electromagnet  (m). 

63 


Sometimes  it  is  more  convenient  to  make  use  of  a 
spring,  by  means  of  which  the  coil  may  be  removed 
from  the  field  with  the  desired  speed.  In  either  case 
the  interval  of  time  should  be  comparable  with  that 
necessary  to  turn  the  earth  inductor  through  180°, 
otherwise  the  impedance  of  the  circuit  will  not  be  quite 
the  same  for  the  two  operations.  A  more  important 
matter  is  that  of  the  placing  of  the  coil  within  the  field. 
The  distribution  of  lines  of  force  in  the  fields  of  the 
character  of  those  for  the  exploration  of  which  the  de- 
vice under  consideration  is  applied  is  by  no  means  uni- 
form. The  number  of  lines  which  will  penetrate  the 
coil  before  it  is  taken  from  the  field  in  successive  trials, 


FIG.  38. 


will  often  be  found  to  vary  considerably  unless  the 
greatest  care  is  taken  to  bring  it  each  time  to  precisely 
the  same  position. 

In  carrying  out  this  method  it  will  frequently  be 
found  necessary  to  vary  the  resistance  R  so  as  to  render 
the  deflections,  due  to  the  motion  of  the  earth  inductor 
and  of  the  small  coil,  comparable.  We  have  then  in  the 
computation  of  the  ratio  of  the  fields  H^.  and  Zfe,  two 
areas  (those  of  the  respective  coils)  Ax  and  Avt  and  two 
resistances  72X  and  Ee. 

Equation  86  takes  the  following  form  when  modified 


64 


to  express  the  relations  existing  in  this  application  of 
the  method,  viz.: 


Sill 


2  R^  Ae  sl 


(96) 


G 

0 


-       f 

6 — ' 


FIG.   39. 

It  is  in  most  cases  convenient  to  place  the  coils  Ax  and 
Ae  in  series  with  one  another.  The  arrangement  of 
connections  is  that  shown  in  Fig.  39. 

Where  the  comparison  of  fields  is  for  the  purpose  of 


o 

.174' 


o 

.1728 


o 
.1750 


o 

.1748 


o 

1750 


o 

.18.9 


o 

.1836 


o 

.1809 


o 

..805 


o 

o 

O 

o 

.-363 

•'553 

•'349 

.1600 

o 

O 

0 

.1684 

.188. 

.2772. 

o 

.'653 


o 
1717 


O 
,1730 


O 

.1770 


FRANKLIN  HALL. 


o 

.r685 


FIG.  40. 

determining  H  in  the  locality  where  a  galvanometer  is 
placed  it  is  important  to  make  the  measurement  as  close- 
ly as  possible  for  the  region  occupied  by  the  needle.  It 
will  not  do  to  assume  that  the  value  of  H  throughout 
an  ordinary  laboratory  room  is  nearly  constant.  R.  W. 

65 


Wilson*  has  pointed  out  the  wide  range  in  the  values  of 
H  within  the  limits  of  the  Jefferson  Laboratory  at 
Cambridge,  and  his  experience  has  been  abundantly 
confirmed  elsewhere.  Fig.  40  shows  the  results  of  a 
similar  survey  recently  made  under  the  direction  of  the 
writer  by  Messrs.  C.  E.  Hewitt  and  A.  W.  Smith.  The 
exploration  covered  the  interior  and  surroundings  of 
the  annex  to  Franklin  Hall.  The  latter  building  is  the 
physical  laboratory  of  Cornell  University,  and  the 
annex  is  a  one-storied  brick  structure,  100  feet  X  36  feet, 
situated  a  few  feet  to  the  north  of  the  main  laboratory. 
It  contains  but  little  iron  aside  from  gas  and  steam 
pipes,  some  cast-iron  wall  brackets  and  some  rods  which 
serve  to  strengthen  the  roof. 

In  the  accompanying  map,  Fig.  40,  A  is  the  annex, 
and  the  various  small  circles  are  stations  at  which  H 
was  determined.  The  values  in  c.  G.  s.  units  for  each 
station  are  indicated.  It  will  be  seen  that  within  the 
building  77  varied  between  .1363  and  .2772,  .1728  being 
the  normal  value  in  localities  distant  from  local  sources 
of  disturbance. 

It  is  interesting  to  note  that  the  stations  just  north  of 
the  annex,  also  those  along  the  north  wall  of  Franklin 
Hall  show  values  of  H  above  the  normal,  while  the  row 
of  stations  just  within  the  north  wall  all  have  low 
values.  A  similar  survey  of  the  neighborhood  of  the 
Magnetic  Observatory  of  Cornell  University,  made  by 
F.  J.  Rogers,  shows  the  same  phenomenon,  and  it  seems 
probable  that  walls  of  masonry  always  exert  a  magnetic 
influence  of  the  kind  described. 

*  R.  W.  Wilson:  Am.rican  Journal  of Science •,  vol.  39.  p.  87.  1890. 


66 


LECTURE  VI. 

THE  CONSTRUCTION  OF  GALVANOMETERS  OF  EXTREME 
SENSITIVENESS. 

There  is  probably  no  instrument  of  precision  used  in 
physics  at  the  present  day  which  possesses  so  wide  a 
range  of  sensitiveness  as  the  galvanometer.  The  ana- 
lytical balance  which  for  a  long  time  was  the  most  re- 
markable of  all  instruments  in  this  respect  has  fallen 
into  second  place  on  account  of  the  remarkable  develop- 
ments as  regards  extreme  sensitiveness  which  have 
been  made  in  the  construction  of  the  galvanometer 
within  a  few  years. 

The  equation  of  the  galvanometer  given  in  the  first 
lecture  indicates  the  lines  along  which  increase  in  sensi- 
tiveness is  to  be  attained.  The  constant  of  the  gal- 
vanometer is  made  up  of  two  parts,  H  the  horizontal 
component  of  the  earth's  magnetism,  and  G  a  factor 
which  depends  upon  the  dimensions  of  the  coil  and  its 
distance  from  the  needle. 

It  is  evident  from  equation  7, 


that  any  method  which  will  increase  the  value  of  G  or 
diminish  H  will  increase  the  sensitiveness  of  the  instru- 
ment. The  latter  of  these  two  processes  has  been  dis- 
cussed at  some  length  in  Lectures  IV.  and  V.,  in  the 
course  of  which  it  has  been  shown  that  an  artifical  field 
may  be  substituted  for  the  earth's  magnetic  field,  this 
field  being  either  stronger  or  weaker  than  the  earth's 
field  according  to  the  purpose  for  which  the  instrument 
is  designed. 

The  final  result  of  weakening  the  field  around  the 
magnet  needle  is  to  produce  greater  and  greater  insta- 
bility of  zero  until  finally  the  drifting  of  the  needle 
becomes  so  rapid  as  to  make  it  impossible  to  obtain 
readings.  Thus  a  limit  to  the  usefulness  of  the  method 
of  weakening  the  field  for  the  purpose  of  increasing 

67 


the  sensitiveness  of  the  instrument  is  reached.  In  many 
operations,  also,  the  lengthening  of  the  period  of  vibra- 
tion would  in  itself  bring  us  to  a  limit  of  usefulness  in- 
jlependent  of  the  matter  of  magnetic  drift. 

Not  less  important  than  the  reduction  of  H  to  small 
values  is  the  increase  of  the  quantity  G  in  the  constant 
of  the  galvanometer;  and  since  this  factor  increases  as 
the  distance  between  the  needle  and  the  wire  diminishes 
and  increases  also  with  the  number  of  turns  of  wire  in 
the  coil,  the  problem  of  construction  with  view  to  ex- 
treme sensitiveness  consists  in  part  of  reducing  to  a 
minimum  the  mean  distance  of  the  windings  from  the 
needle  and  of  getting  the  largest  number  of  complete 
turns  for  a  given  electrical  resistance  in  the  wire  used 
in  the  construction  of  the  instrument.  'A  third,  and 
very  important,  factor  which  enters  into  the  considera- 
tion of  the  construction  of  galvanometers,  is  the  light- 
ness of  the  moving  parts. 

It  is  true  that  the  sensitiveness  of  a  galvanometer 
which  is  used  following  the  method  of  permanent  de- 
flections is  independent  of  the  mass  and  moment  of 
inertia  of  the  moving  parts,  and  independent  of  the 
magnetic  moment  of  the  needle  also.  In  order  to  ren- 
der a  galvanometer,  the  suspended  parts  of  which 
possess  a  large  moment  of  inertia,  as  sensitive  as  one 
in  which  the  moving  parts  are  light,  it  is  necessary, 
however,  to  increase  the  period  of  oscillation;  and  for 
many  purposes  this  consideration  taken  by  itself  would 
dictate  the  reduction  of  the  mass  of  the  suspended  por- 
tions to  a  minimum. 

In  nearly  all  operations  of  extreme  sensitiveness,  gal- 
vanometers are  used  ballistically,  and  under  these  con- 
ditions, both  the  moment  of  inertia  and  the  magnetic 
moment  are  involved  in  the  question  of  sensitiveness. 
The  problem  of  the  maker  of  such  instruments,  there- 
fore, includes  the  question  of  securing  as  large  a  mag- 
netic moment,  and  as  small  a  moment  of  inertia  as 
possible. 

The  sensitive  galvanometer  owes  its  origin  to  the 
demands  of  the  student  of  radiant  energy,  and  it  was 
at  the  hands  of  Nobili  and  of  Melloni  that  two  of  the 
important  steps  toward  increased  sensitiveness  were 
made.  The  first  of  these  was  the  introduction  of  the 
astatic  pair  in  place  of  a  single  needle,  a  device  which 
in  modified  and  refined  form  holds  its  place  in  nearly 
all  modern  instruments.  The  other  step  consisted  in 
the  use  of  the  telescope  and  mirror.  The  galvanometer 
is  a  direct  descendant  of  the  magnetic  compass,  and  the 
user  of  the  galvanometer  inherited  from  his  forerunner, 
the  mariner  who  steered  by  the  aid  of  the  compass,  the 
crude  device  of  a  metallic  pointer  moving  over  a  divided 

68 


circle.  The  substitution  of  the  angular  movement  of  a 
ray  of  light,  noted  by  the  aid  of  the  telescope  and  scale, 
was  a  great  advance. 

The  next  important  step  resulted  from  the  demands 
of  the  needs  of  sub  marine  telegraphy,  and  the  require- 
ments of  this  branch  of  applied  electricity  were  com- 
pletely and  beautifully  met  in  the  mirror  golvanometers 
of  Thomson.  In  these  well-known  instruments  the 
mass  of  the  moving  parts  was  for  the  first  time  reduced 
to  a  small  quantity.  The  needle  which,  at  the  time  of 
Melloni,  was  still  really  a  needle,  taken  without  any 
modification  from  the  hands  of  the  seamstress,  and 
which  in  the  later  galvanometers  of  Siemens,  Wiede- 
mann,  Edelmann  and  others,  had  undergone  a  series  of 
transformations,  none  of  which,  however,  had  been  in 
the  direction  of  diminishing  its  mass  materially,  was 
reduced  by  Kelvin  to  a  system  of  short,  thin  strips  of 
steel,  the  length  of  each  which  was  but  a  few  milli- 
meters, while  the  aggregate  mass  was  a  few  milli- 
grams. 

In  the  hands  of  Kelvin,  also,  the  mirror  was  reduced 
in  weight  in  like  proportion,  by  the  substitution  of 
microscopic  cover  glass  for  polished  metal,  or  for  the 
thick  sheets  of  glass  which  had  been  used  in  the  gal- 
vanometers of  previous  designers.  In  his  instruments 
we  find  also,  for  the  first  time,  the  coil  brought  into 
really  close  proximity  to  the  needles.  The  result  of 
these  changes  was  an  instrument,  the  sensitiveness  of 
which  far  exceeded  that  of  any  instruments  which  had 
previously  existed,  while  the  quickness  of  action  neces- 
sary in  cable  signalling  was  secured  by  the  reduction 
of  the  mass  of  the  moving  parts. 

The  discussion  of  the  proper  form  and  method  of 
winding  galvanometer  coils  to  secure  a  maximum  effect 
from  a  given  weight  or  resistance  of  copper  has  been 
given  by  Maxwell  in  his  treatise  *  The  two  most  im- 
portant points  to  be  considered  are  the  winding  of  the 
coil  with  different  sizes  of  wire  beginning  with  the 
smallest  diameter,  and  the  construction  of  the  coil  in 
such  a  manner  as  to  bring  the  largest  number  of  turns 
within  a  given  effective  distance  from  the  needle. 

If  we  consider  the  action  upon  a  needle  at  JV(Fig.  41) 
of  a  single  turn  of  length  I,  causing  a  current  2,  we 
have  for  the  strength  of  the  component  of  the  magnetic 
field  at  tf,  parallel  to  the  axis  of  the  galvanometer  (see 
equation  3), 

/  --  *  7  sin 
~ 


where   d  is  the   distance  between   the  wire   and  the 
needle. 

*  Electricity  and  Magnetism,  vol.  ii.,  p.  360. 

69 


Since  it  is  upon  this  field  that  the  action  of  the  gal- 
vanometer depends,  it  is  clear  that  the  problem  consists 
in  placing  the  winding,  the  radius  of  which  is  d  sin  6, 
where  it  will  make  a  field  with  the  largest  component 
in  the  required  direction. 

Maxwell  has  shown,  in  the  paragraph  just  cited,  that 
if  a  surface,  the  polar  equation  of  which  is 

d?  =  x?  sin  6.  (98) 

be  constructed,  any  circular  winding  of  length  /  will 
produce  a  greater  effect  when  it  lies  within  the  surface 
than  when  it  lies  outside  it.  It  follows,  therefore,  that 
if  a  completed  coil  be  of  such  shape  that  its  surface  is 
not  of  the  above  form,  we  may  shift  windings  from 


FIG.  41. 


without  the  surface  to  a  position  within  the  same,  thus 
improving  its  action  without  changing  the  amount  of 
wire  used.  In  a  word,  each  layer  of  an  ideal  coil  will 
always  lie  in  a  surface  having  an  equation  of  the  form 
of  (98),  and  the  value  of  x  in  the  expression 


a?  = 


sn 


(99) 


will  be  constant  for  all  its  turns. 

Fig.  42  is  a  diagram  showing  cross-sections  of  three 
such  surfaces  [Maxwell,  ii.,  p.  361]. 

As  regards  the  diameter  of  wire  to  be  used  in  wind- 


ing,  the  chief  results  of  the  discussion,  cited  above,  are 
stated  by  Maxwell,  as  follows: 

i.  "If  the  method  of  covering  the  wire  and  of  wind- 
ing it  is  such  that  the  space  occupied  by  the  metal 
bears  the  same  proportion  to  the  space  between  the 
wires  whether  the  wire  is  thick  or  thin,  then 


Y 

y 


dy 


(100) 


(where  y  is  the  radius  of  the  wire  and  Y9  is  the  area  of 
the  quadrilateral  whose  angles  are  the  sections  of  the 
axes  of  four  neighboring  wires  of  the  coil  by  a  plane 
through  the  axis  of  the  latter),  and  we  must  make  both 


FIG.  42. 


FIG.    43. 


y  and  Y  proportional  to  x  (see  equation  99);  that  is  to 
say,  the  diameter  of  the  wire  in  any  layer  must  be  proportional 
to  the  linear  dimension  of  that  layer'' 

2.  "  If  the  thickness  of  the  insulating  covering  is 
constant  and  equal  to  b,  and  if  the  wires  are  arranged 
in  square  order 

T='2(y  +  b)  (101) 

and  the  condition  is 


(2/+  ft) 


=  constant. 


(102) 


V 

In  this  case  the  diameter  of  the  wire  increases  with 
the  diameter  of  the  layer  of  which  it  forms  a  part,  but 
not  at  so  great  a  rate." 


3-  "  If  increase  of  resistance  is  not  regarded  as  a  de- 
fect, as  when  the  external  resistance  is  far  greater  than 
that  of  the  galvanometer,  or  when  our  only  object  is  to 
produce  a  field  of  intense  force,  we  may  make  y  and  T 
constant.  In  this  case  the  value  of  G  increases  uni- 
formly as  the  dimensions  of  the  coil  are  increased  so 
that  there  is  no  limit  to  the  value  of  G  except  the  labor 
and  expense  of  making  the  coil/'* 

Aside  from  the  questions  of  the  shape  of  coils  and 
the  grading  of  the  wire,  the  construction  of  a  delicate 
galvanometer  depends,  as  has  been  indicated  already, 
upon  the  lightness  of  suspended  parts,  and  the  arrange- 
ment of  same  with  reference  to  a  minimum  value  of 
the  moment  of  inertia,  upon  the  strength  of  the  needles, 
and  upon  the  reduction  of  the  space  within  which  the 
suspended  parts  swing. 

In  all  three  of  these  particulars,  modern  practice 
seems  to  have  been  carried  to  a  definite  limit,  beyond 
which  it  is  difficult  to  proceed.  The  independent  efforts 
of  three  or  four  of  the  most  recent  workers  in  this  field 
have,  indeed,  led  to  the  simultaneous  development  of 
instruments  essentially  identical  and  possessing  very 
nearly  the  same  relative  figure  of  merit. 

The  use  of  miscroscope  cover  glass  for  the  galvano- 
meter mirror  necessitates  the  careful  study  of  the 
materials  used,  since  glass  in  these  thin  layers  is  in- 
variably badly  warped.  One  plan  has  been  to  silver  a 
very  large  number  of  covers,  using  Draper's  solutions 
or  the  rather  more  convenient  ones  recommended  by 
Kohlrausch.  It  is  a  matter  of  great  difficulty  to  find 
among  a  lot  of  mirrors  thus  silvered,  even  one  of  any 
considerable  size  which  presents  a  plane  surface;  but 
fortunately  the  reduction  of  the  face  of  the  mirror  to  a 
minimum  is  a  desirable  thing  where  we  are  seeking  to 
construct  an  instrument  with  very  small  moment  of 
inertia.  One  may  rest  content,  therefore,  with  a  few 
square  millimeters  of  surface,  provided  by  means  of 
these  the  scale  can  be  read. 

Undoubtedly  the  best  procedure  is  that  recommended 
by  Snow,  Franklin  and  others,  which  consists  in  using 
a  glass  plate  with  a  plane  surface  as  a  test  plate  and  of 
laying  down  upon  the  same,  one  after  another,  the 
various  pieces  of  cover  glass  from  which  mirrors  are  to 
be  selected.  If  these  be  properly  cleaned,  interference 
bands  will  show  themselves  and  from  the  shape  of  these 
it  will  be  possible  to  determine  whether  any  portion 
of  the  surface  of  each  is  approximately  plane.  Those 
which  show  the  best  surfaces  are  to  be  laid  aside  and 
silvered;  the  others  are  useless  for  the  making  of  mir- 

*  Maxwell  :  Treatise  ii.,  pp.  363-364. 

72 


U  N  I  v  i: 


rors.  These  selected  glasses  having  been  silvered  in 
the  usual  manner,  should  then  be  cut  into  small  rectan- 
gular pieces  of  the  sizes  desired. 

The  best  size  of  mirror  for  galvanometers  of  the 
highest  sensitiveness  is  the  smallest  size  which  will 
admit  of  readings  being  made  with  the  telescope  and 
scale.  It  is  found  that  when  such  a  mirror  is  cut  to  a 
width  of  less  than  two  millimeters,  diffraction  fringes 
begin  to  disturb  the  image  seriously.  This,  therefore, 
may  be  taken  as  the  limiting  size  of  such  a  mirror. 
We  may  gain  something  in  surface  without  increasing 
the  moment  of  inertia,  appreciably  by  making  the 
mirror  oblong  in  shape  and  mounting  it  with  its  longer 
diameter  parallel  to  the  suspension  rod.  The  best  size 
for  many  purposes  would  seem  to  be  a  length  of  four 
to  five,  with  a  width  from  two  to  two  and  one-half 
millimeters. 

Experience  shows  that  one  of  the  best  materials  for 
the  suspension  bar  or  rod  upon  which  the  elements  of 
the  astatic  pair,  together  with  the  mirror,  are  to  be 
mounted,  is  a  slender  fibre  of  glass.  This  must  be  as 
nearly  straight  as  possible,  since  it  forms  the  axis  of 
rotation  of  the  system.  If  a  considerable  number  of 
glass  fibers  are  made  by  drawing  in  the  flame  and  are 
cut  to  the  proper  length,  the  straight  ones  may  be 
selected  by  laying  all  upon  a  flat  surface  and  rolling 
them  both  back  and  forth  under  the  finger. 

The  question  of  the  best  size  and  shape  for  galvan- 
ometer needles,  where  the  object  is  delicacy,  is  one  upon 
which  some  further  investigations  should  be  made.  At 
present  it  is  generally  conceded  that  a  system  of  three 
or  five  small  needles  arranged  side  by  side,  instead  of 
one  heavier  needle,  gives  a  better  result.  Snow,  in  his 
galvanometer  constructed  in  Berlin  for  the  exploration 
of  the  bright  line  spectra  of  the  metals,  used  six  strips 
in  each  element  of  his  astatic  pair.  These  were  arrang- 
ed pair  wise,  back  to  back,  one  long  between  two  shorter 
pairs.  (Fig.  43.)  Professor  W.  S.  Franklin  and  the 
writer,  in  the  course  of  a  recent  investigation  requiring 
the  very  highest  attainable  sensitiveness  in  the  galvan- 
ometer, made  use  of  an  instrument  in  which  the  needles 
were  prepared  as  described  below.* 

"  The  elements  of  the  astatic  pair  contained  four  mag- 
nets each.  They  were  very  nearly  equal  in  strength, 
and  were  only  two  inches  apart, — the  mirror  being 
placed  below  the  coils  instead  of  between  them.  By 
this  arrangement  the  galvanometer  was  rendered  com- 
paratively insensible  to  magnetic  disturbances.  The 
galvanometer  was  provided  with  freshly  made  magnets 

*  See  Physical  Review,  Vol.  i.  p.  437. 

73 


just  before  being  used, — a  precaution  which  should 
always  be  taken  in  preparing  for  any  important  work 
requiring  the  last  degree  of  sensitiveness,  and  care  was 
taken  to  send  no  currents  of  any  ordinary  strength 
through  the  instrument.  In  the  preparation  of  the 
magnets  the  following  precautions  were  taken.  Piano 
wire  ^  mm.  in  diameter  was  straightened  by  subjecting 
it  to  slight  tension  at  a  low  red  heat,  and  was  cut  into 
two-inch  lengths.  These  were  placed,  two  or  three  at 
a  time,  in  an  acute  V-shaped  iron  trough,  and  atter 
being  heated  uniformly  to  a  cherry-red  heat  (800°  C.) 
in  a  Bunsen  flame,  were  quickly  dropped  into  cool  water. 
Two  small  pieces  of  exactly  the  same  length  were  then 
cut  from  the  central  portion  of  each,  and  magnetized 
under  similar  conditions.  The  cutting  was  done  by 
placing  the  hardened  wire  upon  a  smooth  block  of  hard 
wood,  and  pressing  an  edged  tool  against  it.  If  this 
procedure  be  carefully  followed  a  highly  astatic  pair 
may  always  be  obtained. 

The  two  small  magnets  thus  made  from  each  piece 
were  used,  one  in  each  of  the  elements  of  the  astatic 
system.  The  galvanometer  had  a  resistance  of  15  ohms 
with  its  coils  in  series.  When  so  arranged,  and  with  a 
half -period  of  seven  seconds,  and  scale  distance  of  120 
cm.,  a  deflection  of  i  mm.  corresponded  to  6  x  io~10 
amperes." 

This  refinement  of  the  moving  parts  of  the  galvano- 
meter would  have  been  of  little  use  but  for  the  discov- 
ery of  the  remarkable  qualities  of  quartz  fibres  made 
some  years  ago  by  Professor  C.  V.  Boys.  It  has  been 
abundantly  shown  by  that  physicist,  and  his  statements 
have  been  verified  by  many  others,  that  quartz  pos- 
sesses a  strength  be}rond  that  of  any  other  known  ma- 
terial when  drawn  into  fine  threads  or  fibres,  and  that 
it  is  free  from  the  structural  defects  of  cocoon  silk, 
which  had  been  previously  the  best  of  known  materials 
for  the  suspension  of  galvanometer  needles. 

The  method  of  obtaining  quartz  fibres  described  by 
Boys  is  one  the  execution  of  which  demands  a  con- 
siderable amount  of  manipulative  skill  and  no  little  ex- 
perience. The  following  is  a  more  simple  procedure. 

The  apparatus  needed  is  a  oxyhydrogen  blowpipe, 
two  pairs  of  crucible  tongs  and  some  bits  of  white 
quartz.  The  common  variety  of  quartz  crystal,  known 
as  milky  quartz,  serves  well  for  this  purpose.  The  ma- 
terials should  be  crushed  into  small  granules  about 
three  or  four  millimeters  in  diameter.  These  show  a 
tendency  to  disintegration  when  first  heated  ;  when 
fused,  however,  the  material  goes  over  into  a  condition 
such  that  it  may  be  placed  in  the  flame  over  and  over 

74 


again  after  becoming  cold  without  further  rupture. 
The  first  step  consists  in  making  from  these  bits  of 
pulverized  quartz  a  number  of  short  rods  of  the  molten 
silica.  These  should  be  long  enough  so  that  they  can 
be  held  in  the  flame,  each  end  being  within  the  jaws  of 
a  pair  of  tongs.  When  thus  exposed  to  the  hottest  part 
of  the  gas  jet,  the  middle  softens  readily  and  the  rod 
can  be  drawn  out  into  a  fibre.  These  fibres  are  still 
much  too  heavy  for  use  in  the 
suspension  of  a  delicate  galvano- 
meter, but  they  can  be  reduced  to 
the  desired  fineness  by  the  very 
simple  process  of  holding  them  in 
the  flame  until  they  soften,  when 
the  draught  of  heated  gas  from  the 
nozzle  of  the  burner  will  be  found 
sufficient  to  carry  the  softened 
fibre  with  it,  drawing  it  out  to  a 
thinness  which  renders  it  suitable 
for  the  purpose  now  under  con- 
sideration. With  a  little  care  these 
attenuated  fibres,  which  are  too 
small  to  be  readily  seen,  can  be 
secured,  since  they  are  attached 
at  one  end  to  the  larger  filament 
held  in  the  hand.  It  is  not  al- 
ways possible  to  secure  the  fibres 
in  this  way,  many  of  them  being 
torn  loose  and  swept  away  in  the 
currents  of  air.  By  placing  at  a 
safe  distance  above  the  flame  a 
piece  of  canton  flannel,  however, 
these  stray  fibres  will  be  driven 
against  the  rough  surface  of  the 
cloth  and  will  become  entangled. 
In  a  very  short  time  hundreds  of 
them  may  be  collected  in  this  man- 
ner over  the  oxyhydrogen  flame. 
Many  of  these  will  be  of  consider- 
able length,  and  nearly  all  of  them 
will  be  of  sufficient  fineness  for 
use  in  galvanometers  of  the  high- 
est delicacy. 

The  advantage  of  the  astatic  pair  in  the  construction 
of  galvanometers  having  been  once  recognized,  it  was 
a  very  natural  extension  of  the  principle  to  introduce  a 
second  set  of  coils,  so  that  each  element  of  the  pair 
might  be  brought  into  a  stronger  field  due  to  the  cur 
rent.  Kelvin  made  use  of  such  an  arrangement  in  one 
of  his  types  of  galvanometer,  bringing  the  mirror  to  a 


FIG.  44. 


75 


position  midway  between  the  two  groups  of  needles,  a 
procedure  which  has  been  widely  followed  by  others. 
Fig.  44  shows  an  instrument  in  which  four  coils  are  used 
with  the  mirror  placed  between  them  as  above  de- 
scribed. Fig.  45  shows  the  same  instrument  on  a 
larger  scale  with  the  short  coils  swung  away  so  as  to 
show  the  interior.  Fig.  46  shows  the  arrangement  of 


FIG.  45. 


the  needles  and  mirror  in  this  instrument   about  life 
size. 

The  suspended  parts  of  such  galvanometers  having 
been  reduced  to  a  minimum  as  regards  mass  and  moment 
of  inertia,  it  was  a  natural  mistake  to  suppose  that  even 
the  quartz  fibre  must  be  of  great  length  in  order  that 
its  moment  of  torsion  should  remain  inappreciable. 
Thus  in  the  galvanometer  just  depicted  a  fibre  half  a 


meter  long  was  used,  and  of  many  other  galvanometers 
made  at  that  time  the  same  thing  is  true.  While  it  is 
easy  to  obtain  quartz  fibres  of  the  requisite  length  and 
fineness,  it  is  a  much  more  serious  matter  to  mount  a 
long  fibre  than  a  short  one,  and  after  the  instrument 
has  been  successfully  set  up,  the  question  of  keeping  it 
adjusted,  as  to  level,  so  that  the  suspended  parts  shall 
be  free  within  the  very  narrow  space  allotted  to  them, 
becomes  a  difficult  one.  Subsequent  experience  has 
shown  that  a  fibre  five  to  ten  centimeters  long  is 
sufficient,  even  in  the  case  of  the  lightest  galvanometers. 
There  are  certain  advantages  in  bringing  the  elements 


II 


FIG.    46. 


a  b 

FIG.  47. 


of  an  astatic  pair  near  to  one  another,  and  to  attain  this 
end,  the  mirror  is  sometimes  placed  at  the  bottom  of  the 
suspension  rods  as  shown  in  Fig.  47  (a)  which  gives 
.the  arrangement  of  mirror  and  needles  in  such  a  gal- 
vanometer. Fig.  48  shows  an  instmment  of  this 
type,  constructed  as  were  also  the  galvanometers  shown 
in  Fig.  44  and  49,  by  F.  C.  Fowler,  instrument  maker 
to  the  Department  of  Physics  in  Cornell  University. 
This  instrument  has  four  coils  placed  pair- wise  one 
above  another  and  adjustable  as  to  the  distance  between 
them.  A  somewhat  similar  instrument  with  eight  coils 
and  four  sets  of  needles  mounted  in  four  equally  distant 


77 


groups  with  a  mirror  at  the  bottom  is  shown  in  Fig. 
49.  A  diagram  of  the  suspended  parts  is  given  in  Fig. 
47  (b)  Figures  50,  51  and  52  give  some  details  of  the 
galvanometer  shown  in  Fig.  49.  , 

The  only  factor  in  the  construction  of  a  sensitive  gal- 
vanometer, which  we  have  still  to  consider,  is  that  of 
the  distance  between  the  opposite  coils.  With  mirrors 
two  millimeters  across  and  needles  three  or  four  milli- 
meters long,  it  would  seem  that  the  clearance  necessary 


FIG.    48. 


to  give  freedom  of  action  might  be  extremely  small. 
In  this  regard,  however,  a  limit  is  soon  reached  on 
account  of  the  difficulties  which  arise  through  the  mag- 
netic properties  of  the  materials  of  which  the  coils 
themselves  are  constructed.  If  the  layers  of  wire  lying 
next  the  needle  be  insulated  with  the  usual  green  silk, 
it  will  be  found  that  these  when  brought  near  the  sus- 
pended needles  will  attract  the  same  strongly  and  will 


soon  interfere  altogether  with  the  requisite  freedom  of 
motion.  The  substitution  of  white  for  the  green  cover- 
ed wire  seems  to  mitigate  this  trouble  to  some  degree, 
but  it  is  a  matter  of  great  difficulty  to  find  thoroughly 
non-magnetic  insulated  copper  wire.  Then,  again,  in 
the  process  of  handling  the  instrument  for  mounting, 
the  silk  covering,  and  the  shellaced  surfaces,  tend  to 
become  electrified  and  much  annoyance  arises  from  this 
source.  It  is,  indeed,  sometimes  necessary  to  cover  the 


FIG.    49. 

entire  inner  face  of  the  coils  with  gold  leaf  and  to 
ground  the  same  before  the  needles  can  be  made  to 
swing  freely  in  close  proximity  to  the  coils.  The  writer 
has  never  found  it  practicable  to  work  with  galvano- 
meters in  which  the  average  clearance  space  was  re- 
duced to  less  than  one  millimeter.  Fig.  53,  which  shows 
the  position  of  the  coils  of  Snow's  galvanometer,  in- 
dicates that  in  his  instrument,  the  clearance  was  approx- 
imately that  just  mentioned. 


79 


As  Regards  the  sensitiveness  of  galvanometers  con- 
structed in  accordance  with  the  principles  laid  down  in 
this  lecture,  the  following  data  may  be  of  interest: 

B.  W.  Snow,  1892,  constructed  the  galvanometer,  the 
coils  of  which  are  shown  in  Fig.  53,  and  the  suspended 
parts  in  Fig.  43.  He  obtained  a  figure  of  merit  1.5  X  io~n 
amperes,  with  a  vibration  period  of  20  seconds  and  a 
scale  300  cm.  from  the  instrument.  The  above  applies 
to  a  deflection  of  one  millimeter.  The  resistance  of  the 


FIG.  50. 


FIG.  51. 


FIG.    52. 


instrument  was  140  ohms.  In  the  same  year  E.  F. 
Nichols  and  the  writer  used  the  galvanometer  with  long 
suspension  fibre,  shown  in  Fig.  44.  The  aggregate 
weight  of  the  moving  parts  of  this  instrument  was 
48  mg.  With  the  coils  in  multiple  the  resistance  was 
9.3  ohms.  The  sensitiveness  reached  i  X  icr10  amperes 
with  the  coils  thus  connected.  The  scale  was  about  150 
cm.  from  the  instrument,  and  the  period  about  10 
seconds.  W.  S.  Franklin  and  the  writer  found  for  the 


80 


galvanometer  used  in  their  study  of  the  condition  of 
the  ether  surrounding  a  moving  body,  the  method  of 
constructing  the  magnets  of  which  instrument  has  just 
been  described,  a  figure  of  merit  of  6  x  io~10  amperes 
with  a  resistance  of  150  ohms,  a  period  of  7  seconds  and 
a  scale  of  1 20  cm.  distant.  In  this  case  also  the  figure 
of  merit  refers  to  a  millimeter  of  deflection.* 

Paschenf  also,  who    constructed    a   special   galvano- 


0 


o 


FIG.  53 


meter  for  the  exploration  of  the  very  weak  spectra 
afforded  by  the  diffraction  grating,  obtained  i  6  X  io~n 
amperes  for  one  millimeter  with  20  ohms,  a  period  of 
30  seconds  and  a  scale  270  cm.  distant. 


*  For  these  three  cases  see  Physical  Review,  vol.  i. 
t  Paschen :   Wiedemanrfs  Annalen,  48,  p.  284. 


LECTURE  VII. 

SPECIAL  APPLICATIONS  OF  THE  GALVANOMETER    TO   THE 
MEASUREMENT  OF  CURRENT  AND  RESISTANCE. 

i.  Measurement  of  feeble  currents. — One  of  the  import- 
ant uses  of  the  galvanometer  is  for  the  detection  and 
measurement  of  currents  of  exceedingly  small  intensity, 
for  which  purpose  the  instrument  must  be  especially 
adapted  by  constructing  it  with  reference  to  the  reduc- 
tion of  the  constant  to  the  smallest  possible  value. 
This  reduction,  as  has  already  been  pointed  out  (Lecture 
VI.),  is  attained  by  bringing  the  wire  as  near  as  pos- 
sible to  the  needle,  by  reducing  the  moment  of  inertia 
of  the  moving  parts  to  a  minimum,  by  making  use  of 
the  astatic  system  and  by  the  artificial  reduction  of  the 
magnetic  field  within  which  the  needle  swings  to  a  very 
small  intensity. 

Since  the  constant  of  such  instruments  cannot  be 
determined  by  computation  from  the  dimensions  of  the 
coils  the  galvanometer  must,  be  calibrated.  In  ad- 
dition to  the  absolute  calibration,  means  must  be  de- 
vised for  repeated  re-determinations  of  the  fluctuations 
to  which  the  figure  ot  merit  of  galvanometers  of  extreme 
delicacy  are  subject. 

The  calibration  may  be  made: 

(i.)  By  the  use  of  the  Clark's  cell,  the  greatest  care 
being  taken  to  fulfill  the  conditions  under  which  this 
form  of  cell  affords  reliable  results.  A  detailed  de- 
scription of  this  cell  and  of  the  method  of  using  it  will 
be  given  in  the  Lecture  VIII. 

(2.)  The  galvanometer  may  be  calibrated  also  by 
placing  it  in  shunt  around  a  known  resistance  (see  Fig. 
54).  This  method  involves  the  use  of  a  second  galvano- 
meter, G,  of  known  constant,  by  means  of  which  the 
current  flowing  through  the  resistance  in  question  can 
be  measured.  The  size  of  this  resistance,  RS,  will  de- 

82 


pend  upon  the  figure  of  merit  of  the  instrument.  It 
must  be  of  such  size  that  when  a  measurable  current 
traverses  it,  the  difference  of  potential  between  its  ter- 
minals, shall  give  a  suitable  deflection  to  the  galvano- 
meter to  be  tested.  In  case  of  instruments  of  the 
highest  delicacy  this  involves  the%  reduction  of  the  cur- 
rent to  values  too  small  to  be  measured  upon  a  stand- 


FIG.    54. 

ard  instrument,  or  the  increase  of  the  resistance  to  such 
an  extent  as  to  render  accurate  knowledge  of  its  value, 
a  matter  of  difficulty.  In  such  cases  it  is  better  to  use 
a  multiple  shunt,  the  arrangement  of  which  is  shown  in 

Fig-  55- 

This  device  makes  it  possible  to  standardize  instru- 
ments of  extreme  sensitiveness  with  a  fair  degree  of 
accuracy.  It  is  true  that  increased  error  is  introduced 
by  the  use  of  the  second  shunt,  but  it  is  also  true  that 


FIG. 


very  precise  determinations  of  the  figure  of  merit  of 
galvanometers  of  the  highest  delicacy  are  rendered  use- 
less from  the  fact  that  such  galvanometers  are  subject 
to  continual  changes  of  constant.  Almost  in  proportion 
as  the  sensitiveness  of  the  instrument  rises  beyond  a 
certain  value,  the  possibility  of  precise  determination 
of  its  sensitiveness  diminishes. 


Arrangements  for  the  calibration  of  the*  instrument 
having  been  completed,  it  is  necessary  to  provide  some 
means  of  keeping  pace  with  the  fluctuations  of  constant 
already  referred  to.  Probably  the  very  best  means  for 
this  purpose  is  a  subsidiary  coil  placed  at  a  suitable  dis- 
tance behind  the  galvanometer.  This  coil  will  act  upon 
the  needle,  but  the  current  through  it  necessary  to  pro- 
duce a  deflection  will  be  very  much  larger  than  that 
which  would  give  the  same  result  when  sent  through 
the  coils  of  the  galvanometer  itself.  The  number  of 
turns  of  this  subsidiary  coil  may  be  few,  but  its  dis- 
tance from  the  needle  is  necessarily  very  considerable. 

The  subsidiary  coil  should  be  placed  in  shunt  circuit 


FIG.    56. 


with  a  compensated  resistance,  rs,  Fig.  56;  through 
which  a  known  current  flows.  Upon  closing  this  sub- 
sidiary circuit  a  deflection  will  be  produced  by  the 
proper  adjustment  of  the  resistance  in  the  circuit.  The 
deflection  may  be  brought  to  size  approximately  equal 
to  that  of  the  deflections  which  the  galvanometer  will 
give  in  the  operations  to  which  it  is  to  be  subjected  by 
further  adjustment  of  the  resistance  r?.  Immediately 
after  the  completion  of  the  absolute  calibration,  the  de- 
flection due  to  the  subsidiary  coil  with  known  resistance 
flowing  through  the  compensated  resistance  should  be 
noted,  and  this  deflection  be  made  to  serve  as  a  refer- 


ence  factor  in  all  subsequent  operations.  In  order  to 
keep  track  of  changes  in  the  constant  of  the  galvano- 
meter occurring  from  time  to  time  thereafter,  it  will 
only  be  necessary  to  send  the  same  current  through  the 
subsidiary  coil  as  that  which  was  sent  through  it  at 
the  time  of  the  calibration.  The  range  in  the  deflec- 
tions thus  produced  will  then  represent  the  range  of  the 
figure  of  merit  of  the  galvanometer.-  The  arrangement 
of  the  subsidiary  coil  and  its  circuit  is  shown  in  Fig. 
56,  to  which  reference  has  just  been  made. 

Another  device  for  obtaining  the  same  end,  which  has 
the  advantage  of  not  requiring  the  use  of  a  standard 
instrument  or  ammeter,  is  as  follows  : 

A  thermo-element  consisting  of  an  iron  or  German 
silver  wire  about  one  meter  long,  the  ends  of  which  are 


FIG.  57. 


soldered  to  copper  wires,  is  constructed.  One  junction 
of  this  thermo-element  is  packed  in  ice,  the  other  is 
placed  in  a  steam  bath,  the  pressure  of  which  can  be 
regulated  so  as  to  make  a  delicate  thermometer  situated 
in  the  same  bath  read  constantly  100  degrees.  Such  a 
thermo-element  will  give  a  small  but  perfectly  constant 
electromotive  force  so  long  as  a  constant  difference  of 
temperature  between  its  terminals  is  maintained.  This 
thermo-element  may  be  used  two  ways:  (i)  to  produce 
deflections  by  placing  it  in  circuit  with  the  galvano- 
meter coils  themselves  from  time  to  time  (see  Fig.  58); 
(2)  in  the  case  of  galvanometers  of  extreme  sensitive- 
ness, by  placing  the  thermo-element  in  circuit  with  the 
subsidiary  coil. 

In  all  cases  in  which  it  is  necessary  to  reduce  the  gal- 
vanometer to  its  condition  of  maximum  sensitiveness,  a 

85 


point  is  reached  at  which  the  drift  of  the  zero  due  to 
magnetic  disturbances  introduces  serious  error.  It  has 
already  been  shown  in  the  lecture  on  galvanometers 
with  artificial  fields,  that  in  the  case  of  instruments 
with  weak  fields  every  fluctuation  in  strength  of  the 
earth's  magnetic  forces  has  an  exaggerated  effect  upon 
the  needle.  It  is,  indeed,  oftentimes  impossible  to 
maintain  a  sensitive  galvanometer  with  approximately 
fixed  zero  long  enough  to  obtain  a  permanent  deflec- 
tion. 

In  all  such  cases  the  only  remedy  is  to  make  use  of 
the  ballistic  method  in  which  the  zero  of  the  galvano- 
meter at  the  instant  before  closing  circuit  is  noted,  and 
the  circuit  is  closed  during  a  single  swing  of  the  instru- 
ment. It  is  then  opened,  the  reading  at  the  end  of  this 


FIG.  58. 


swing  is  noted,  and  finally  the  point  which  the  needle 
reaches  in  its  first  return  swing  is  observed.  If  the 
average  between  the  reading  of  the  return  swing  and 
the  original  zero  be  taken  as  a  corrected  zero,  the  drift 
of  the  galvanometer  in  the  short  intervening  interval 
will  be  almost  entirely  eliminated.  Deflections  com- 
puted in  this  way  should  be  interpreted  by  means  of  a 
calibration  performed  in  a  corresponding  manner  in- 
stead of  a  calibration  by  permanent  deflections. 

In  some  extreme  cases,  such  as  occur  in  work  with 
the  thermo-pile  or  bolometer,  it  is  found  to  be  impos- 
sible to  open  and  close  a  switch  in  the  galvanometer 
circuit  at  all  without  producing  disturbances  of  great 
magnitude.  In  these  cases  fortunately,  it  will  be  found 
possible  to  use  the  galvanometer  in  a  permanently 

86 


closed  circuit,  the  deflection  being  produced  by  the  ex- 
posure of  the  thermo-pile  or  bolometer,  to  the  source 
of  radiation  under  observation,  by  the  sudden  removal 
of  an  intervening  screen. 

The  operation  consists  in  reading  the  instrument  with 
the  circuit  closed,  and  with  the  screen  intervening  be- 
tween the  source  of  light  and  the  thermo-pile  or  bolo- 
meter. This  gives  the  zero  point  of  the  deflection. 
The  screen  is  then  removed  during  one  swing  of  the 
galvanometer  and  is  restored  to  its  place.  The  extreme 
reading  of  the  swing  is  read,  also  the  return  swing,  and 
the  deflection  is  computed  from  these  three  observations 
as  already  indicated.  Theory  would  lead  us  to  expect 
that  the  deflection  thus  obtained  would  be  proportional 


3* 


FIG.    59. 


to  the  permanent  deflection  of  a  galvanometer,  the 
zero  of  which  was  fixed.  Professor  Ernest  Merritt* 
has  shown  experimentally  that  these  conditions  are  ful- 
filled in  practice.  Fig.  59  gives  a  curve  obtained  by 
him  from  observation  of  a  galvanometer  used  in  this 
manner  on  -closed  circuit,  the  readings  being  carried  on 
not  only  during  the  first  swing,  but  for  a  much  longer 
period,  up  to  the  time,  indeed,  when  the  needle  reached 
its  final  position. 

In  this  figure  curve  I  gives  the  observed  movement 
of  the  needle,  the  galvanometer  being  in  closed  circuit 
with  a  thermo  pile  which  was  suddenly  exposed  to  heat 
at  the  time,  marked  O  seconds.  The  analysis  of  this 

*  Merritt ;  American  Journal  of  Science,  vol.  xli.,  p.  417. 


curve  shows  it  to  be  the  resultant  of  curves  II  and  III, 
the  latter,  of  which  is  a  trace  of  logarithmic  decrements 
the  period  of  which  agrees  with  the  free  swing  of  the 
needle. 

Merritt  found  that  in  all  cases  the  first  throw  of  the  gal- 
ganometer  needle  was  proportional  to  its  permanent 
deflection. 

2.  Measurement  of  heavy  currents. — Another  important 
problem  in  the  measurement  of  currents  is  that  where 
the  strength  of  the  current  is  too  great  for  direct  deter- 
minations. Under  these  circumstances  the  method  by 
fall  of  potential  is  to  be  preferred.  The  only  difficulty 
of  carrying  out  this  method  successfully  lies  in  the 
maintenance  of  a  constant  temperature  in  the  resist- 


0.0030 

t 

1 

1 

0.0025 

* 

» 

1 

0.0020 

t 

\ 

V 

1 

0.0015 

1 

\ 

X 

\ 

OOU10 

1 

\ 

V 

1 

\ 

0.0005 

\ 

X 

V 

24 


6 

FIG.    60. 


ance  around  which  the  galvanometer  is  shunted,  but 
this  difficulty  has  been  found  so  great  as  to  lead  to  the 
condemnation  of  the  method  on  the  part  of  many  in- 
vestigators. Two  remedies  have  been  proposed:  (i) 
the  use  of  a  material  for  the  shunt  resistance  which 
does  not  vary  with  the  temperature.  Certain  alloys  of 
manganese  and  copper,  or  manganese,  nicKel  and  copper, 
are  known  to  possess  this  property,  but  their  other  pro- 
perties have  not  been  studied  as  yet  with  sufficient 
thoroughness  to  warrant  unreserved  confidence  in  their 
stability.  In  a  word,  we  do  not  know  whether  shunts 
constructed  of  such  material  will  remain  of  constant 
resistance  when  subjected  to  the  action  of  heavy  cur- 
rents. 

88 


From  the  measurements  made  upon  this  class  of 
alloys,  however,  it  appears  that  unless  very  particular 
attention  is  paid  to  the  matter  of  annealing,  etc., 
changes  of  specific  resistances  are  sure  to  occur  as  the 
result  of  every  subsequent  fluctuation  of  temperature. 
B.  H.  Blood*  in  the  course  of  an  examination  of  an 
alloy  containing  .8082  copper  and  .1912  ferro-manganese, 
found  that  eight  successive  heatings  to  ico°  C.  with 
alternate  coolings  to  20°  C.,  gave  the  results  indicated 
in  the  following  table. 


EFFECT  OF  REPEATED  HEATING  AND  COOLING  UPON  THE 
RESISTANCE  OF  AN  ALLOY  OF  FERRO-MANGANESE  AND 
COPPER  (HARD  DRAWN). 


Observation. 

Temperature. 

Specific  Resistance. 

Relative  Resistance. 

Degrees. 

20 

30.380 

1.  0000 

100 

30.186 

•99331. 

20 

30.163 

.99287 

100 

30.151 

•99255 

20 

30.138 

.99202 

6 

100 

30.121 

.99180 

7 

20 

30.118 

•99134 

8 

100 

30.118 

•99I34 

9 

20 

30.105  • 

.99093 

10 

100 

30.099 

.99072 

n 

2O 

30.092 

.99051 

12 

100 

30.104 

.99092 

13 

20 

30.079 

.99007 

*4 

100 

30.104 

.99092 

15 

20 

30.072 

.98985 

As  regards  temperature  coefficients  of  resistance, 
however,  provided  a  method  of  treatment  can  be  found 
to  check  the  behavior  just  referred  to,  this  class  of  al- 
loys leaves  little  to  be  desired.  The  coefficient,  indeed, 
seems  to  depend  directly  upon  of  the  percentage  of  ferro- 
manganese  present,  and  to  pass  from  positive  to  negative 
values  when  18  per  cent,  of  that  material  is  combined 
with  the  copper.  This  is  shown  in  the  curve  (Fig.  60) 
platted  from  results  obtained  by  the  same  observer.  In 
this  diagram  ordinates  are  coefficients  and  abscissae  are 
percentages  of  ferro-manganese. 

Since  the  currents,  the  measurement  of  which  we  are 
now  considering,  are  too  heavy  to  allow  the  use  of  the 
compensated  resistance  of  copper  and  carbon,  which  is 
to  be  described  in  Lecture  VIII.,  and  since  the  only 
known  available  material  which  is  without  a  coefficient 
is  questioned,  recourse  must  be  had  to  a  metallic  shunt, 
and  to  some  device  by  means  of  which  its  temperature 
can  be  controlled.  This,  indeed,  is  the  plan  prescribed 

*  American  Journal  of  Science^  vol.  xxxix.,  p,  473. 

8Q 


in  what  is  known  as  the  Vienna  method.  The  use  of  a 
bath  has  been  found  to  introduce  such  errors,  however, 
as  to  lead  to  the  very  general  abandonment  of  the 
method. 

The  principal  source  of  error  is  that  due  to  the  as- 
sumption of  a  false  value  for  the  resistance  of  the 
metallic  shunt.  The  resistance  is,  of  course,  a  function 
of  the  temperature  of  the  metal,  and  since  a  conductor 
carrying  a  current  beyond  its  normal  capacity  and  de- 
veloping a  large  amount  of  heat  energy  must  necessar- 
ily possess  a  temperature  considerably  above  that  of  its 
surroundings,  it  is  not  allowable  to  assume  that  a 
thermometer  placed  in  a  bath  will  indicate  the  true 
temperature  of  the  shunt. 


FIG.    6l. 

To  make  the  method  a  reliable  one,  it  is  before  all 
necessary  to  determine  the  precise  resistance  of  the 
shunt  for  all  the  conditions  under  which  it  is  to  be  used. 
This  may  be  accomplished  in  one  of  the  folio  wing  ways: 

(i.)  The  shunt  resistance  may  be  constructed  in  such 
a  form  that  it  can  be  readily  surrounded  by  a  coil  of 
fine  insulated  copper  wire  fitting  snugly  to  its  surface 
(see  Fig.  61.)  The  temperature  coefficient  of  the  resist- 
ance of  this  copper  coil  may  be  determined  once  for  all. 
Its  resistance  will  indicate  much  more  closely  than  can 
be  done  by  means  of  a  mercury  thermometer  inserted 
in  a  bath,  the  temperature  of  the  conductor  around 
which  it  is  wound.  This  method  of  measuring  tem- 
peratures is  of  sufficient  delicacy;  it  gives  an  integration 
of  the  temperature  for  the  entire  surface  of  the  shunt 
resistance,  and  it  is,  indeed,  open  only  to  a  single 

90 


objection;  that  the  temperature  measured  is  that  of  the 
region  lying  just  without  the  surface  of  the  metal,  which, 
in  the  cases  under  consideration,  will  always  be  slightly 
lower  than  the  average  temperature  within  the  mass. 
This  error,  which  is  a  small  one,  may  be  further  reduc- 
ed by  using  thin  strips  of  metal  for  the  shunt  resistance 
so  that  the  temperature  differences  between  the  surface 
and  the  interior  shall  be  extremely  small.  Another, 
and  still  more  effectual,  method  of  reducing  the  error 
of  difference  between  the  temperature  indicated  by  the 
calibrating  coil  and  that  which  it  is  desired  to  know, 
consists  in  making  the  shunt  resistance  in  the  form 
of  a  tube,  or  series  of  tubes,  around  which  the  cali- 
brating coil  is  wound  longitudinally  so  that  one-half  its 
length  will  lie  within  the  core  of  the  tube,  in  which 
position  it  will  reach  a  temperature  very  nearly  as  high 
as  that  of  the  mass  of  metal  itself. 

(2.)  The  other  method  of  determining  the  true  tem- 
perature of  the  shunt-resistance  is  free  from  the  errors 
which  have  just  been  pointed  out.  This  method  consists 
in  studying  the  temperature  of  the  shunt  as  a  function 
of  the  time,  a  time  curve  being  taken  immediately  after 
the  close  of  each  measurement  of  current.  The  best 
apparatus  for  getting  this  curve  is  a  calibrating  coil  of 
the  kind  described  under  method  (i.)  The  procedure 
is  as  follows: 

The  current  to  be  measured  having  been  sent  through 
the  shunt  resistance  for  a  sufficient  amount  of  time  to 
allow  the  latter  to  reach  its  final  temperature,  and 
the  readings  by  fall  of  potential  having  been  made, 
the  circuit  is  broken  at  an  instant  of  time  accurately 
noted,  and  a  succession  of  measurements  of  the  resist- 
ance of  the  calibrating  coil  are  made  at  times  likewise 
carefully  observed.  From  the  moment  of  breaking  the 
circuit,  the  shunt  resistance  will  begin  to  cool,  approach- 
ing gradually  to  the  temperature  of  its  surroundings. 
The  measurements  of  resistance  and  time  just  described 
will  permit  of  the  platting  of  a  curve  in  which  abscissas 
are  times  and  ordinates  are  temperatures.  By  extend- 
ing this  curve  backwards  to  the  origin  of  time,  it  is 
possible,  through  extrapolation,  to  obtain  a  very  accu- 
rate determination  of  the  temperature  of  the  shunt  at 
the  instant  when  the  current  ceased  to  flow.  • 

Carried  out  in  this  way,  the  Vienna  method  affords 
an  entirely  reliable  means  of  measuring  heavy  currents. 
Its  accuracy  depends  only  upon  the  precision  with 
which  the  absolute  resistance  of  the  shunt  can  be  deter- 
mined. By  means  of  the  second  method  just  outlined, 
the  change  of  resistance,  which  the  shunt  undergoes  on 
account  of  the  heating  effect  of  the  current  which 
traverses  it,  can  be  accurately  ascertained. 

91 


LECTURE   VIII. 

THE  MEASUREMENT  OF  ELECTROMOTIVE  FORCE  BY  MEANS 
OF  THE  GALVANOMETER. 

The  Determination  of  the  Electromotive  Force  in  Absolute 
Measure. — The  best  instrument  for  this  practice  is  a  tan- 
gent galvanometer  of  considerable  sensitiveness,  the  con- 
stant of  which  has  been  determined  by  measurement  of 
the  coils.  The  resistance  of  the  galvanometer  itself  should 
be  as  low  as  possible,  and  it  should  be  used  in  series 
with  a  variable  resistance,  the  absolute  values  of  all  the 
coils  of  which  are  accurately  known.  Fig.  62  shows  the 
arrangement  of  this  apparatus  for  the  determination  of 
the  difference  of  potential  between  points  a  and  b  in  any 
circuit. 

The  method  is  capable  of  very  wide  range.  The  up- 
per limit  is  reached  when  the  electromotive  force  to  be 
measured  is  so  great  that  the  resistance  RP  cannot  be 
made  large  enough  to  reduce  the  deflection  to  a  read- 
able size.  The  lower  limit  is  reached  when  RP  becomes 
zero  and  the  deflections  are  too  small  for  accuracy. 
The  method  may  be  extended  in  the  direction  of  high 
electro-motive  forces  by  the  use  of  a  galvanometer  with 
two  coils  capable  of  being  placed  in  series,  or  in  mul- 
tiple, or  of  being  used  differentially. 

Fig.  63  is  from  a  photograph  of  a  small  tangent  gal- 
vanometer designed  especially  for  this  purpose  by  Prof. 
W.  A.  Anthony.  The  windings  are  arranged  in  four 
parts  which  may  be  thrown  together  in  either  of  the 
three  ways  indicated  above,  by  the  use  of  a  small  divided 
block  with  plugs  placed  upon  the  base  of  the  instrument. 
The  range  of  usefulness  is  very  wide,  extending  from 
a  thousandth  of  a  volt  per  centimeter  deflection  with  no 
resistance  in  circuit,  to  over  twenty  volts  per  centimeter 
deflection,  when  used  in  series  with  a  megohm  box. 
Used  differentially,  the  instrument  extends  to  quite  as 
high  potentials  as  one  could  expect  to  measure  by  such 
a  method. 

92 


In  a  case  of  very  low  electromotive  forces,  the  stand- 
ard galvanometer  must  be  replaced  by  one  of  high 
sensitiveness,  which  it  is  necessary  to  calibrate  before 
using.  Probably  the  best  method  of  calibrating  such 
a  galvanometer  consists  in  looping  the  same  with  suit- 
able resistance  in  circuit  around  a  compensated  resist- 
ance through  which  a  known  current  flows.  (See  Fig. 
64.)  With  a  suitable  standard  instrument,  A,  for  the 
measurement  of  current  flowing  through  this  compen- 
sated resistance  and  with  adjustable  resistance  RS  and  Rg 
in  the  main  circuit  and  in  the  circuit  of  the  galvano- 
meter to  be  tested,  the  latter  can  be  accurately  calibrated 
throughout  its  entire  range. 

The  compensated  resistance  coil  suitable  for  this 
purpose,  is  made  of  copper,  which,  as  is  well  known, 
possesses  a  positive  temperature  coefficient  of  about 
0.004  placed  in  multiple  with  a  rod  of  graphitic  carbon, 
the  temperature  coefficient  of  which  is  always  negative 
with  a  value  varying  considerably,  but  always  much 
smaller  than  that  of  copper.  (See  Fig.  65.)  Some  data 
concerning  the  variations  of  the  temperature  coefficient 
for  resistance  to  be  met  with  in  testing  various  kinds  of 
carbon  are  given  in  the  following  table. 

INFLUENCE  OF  TEMPERATURE  UPON  THE  RESISTANCE  OF 
VARIOUS  VARIETIES  OF  CARBON. 


Qbserver. 

Variety  of  Carbon. 

Tempera- 
ture 
Interval 

Mean  co- 
efficient 
per  deg. 

Werner     Siemens     (  Wiede-  ( 
ntanns  Annalen,    10,   p.  •< 
56o)                                           1 

Borgmatm      (Journal     der  f 

Gas  carbon  
Pressed  gas  carbon  

Charcoal       

0  tO  200° 

o  to  200° 
26  to  260" 

.000345 
.000301 

.00370 

Anthracite 

20  to  250° 

.00265 

lischen  Gesellschaft,  9,  p.  1 
167  ) 

Graphite  
Coke 

25  to  250° 
26  to  245° 

.00082 
.00026 

Gas  carbon  (coarse)      .... 

18  to  200° 

.000285 

Kemlein  (Wtedemanns  A  n-  j 
nalen,  12,  p.  73.)                    "l 

Gas  carbon  (fine)  
Carry's  carbons       .          .... 

1  8  tO  200° 

18  to  100° 

.000287 
.000321 

f 

Paris  retort  carbons      

.000300 

.000425 

Muroaka     (  Wiedemann  s 

Gaudoin's  carbons  .  .            

.000415 
.000370 

Annalen,  13,  p.  307.) 

.0001  s6 

I 

Siberian  graphite  
Faber's  lead  pencil  graphite.  . 

.000739 
.000588 

For  the  range  of  temperatures  through  which  it  is 
desired  that  this  resistance  shall  be  constant,  we  may 
assume  that  the  coefficients  are  both  constant  quantities. 

The  problem  to  be  solved  in  the  construction    of   a 


93 


compensated  coil  for  the  range  of  temperatures  in  ques- 
tion consists  in  determining  the  proper  amount  of 
carbon  and  of  copper  to  be  placed  in  multiple  circuit  so 
that  the  total  resistance  of  the  combination  shall  not 
vary.  This  condition  will  be  met,  provided  the  rise  in 
resistance  on  the  part  of  the  copper  is  exact.ly  counter- 
balanced by  the  increased  conductivity  of  the  carbon. 
These  conditions  will  be  approximately  fulfilled  when 

«««   )  =  St'  (,-£-?-.),  (103) 

<± — «c  &/  \1  -f-  am  t) 

in  which  equation  Rm'  is  the  resistance  of  copper,  R0' 
that  of  the  carbon  at  a  given  temperature,  (say  2o°C), 
while  am  and  ac  are  the  respective  coefficients  and  t  is  a 
temperature  (say  ioo°C),  up  to  which  compensation  is 
desired. 


FIG.    62. 


Assuming  that  the  combination  would  be  subject  to 
the  law  of  parallel  circuits,  we  have  for  the  total  resist- 
ance R: 


where  Rm  is  the  resistance  of  the  metal  and  Rc  the  re- 
sistance of  the  carbon. 

The  equation  of  condition,  for  complete  compensation 
will  be 

R        B  -  ^ 


where  Rm'  Rc'  are  the  resistances  of  the  components  and 

94 


fim"  -^c*.  the  corresponding  resistances  at  any  other 
temperature  within  the  limits  of  temperature  for  which 
compensation  exists. 

Now  the  variation  of  the  resistance  of  a  metal  with 
the  temperature  may  be  expressed  by  an  equation  of 
the  form  : 

-£m*=-£m'  (1  +  at  ±  W\  (106) 

where  a  and  b  are  coefficients  to  be  determined  by  ex- 
periment. 


FIG.    63. 


In  the  case  of  carbon,  the  coefficient  a  will  have  a 
negative  value,  and  the  equation  will  take  the  following 
form  : 

7?c"  =  Rc'  (i  —  at±  If).  .    (107) 

Between  o°  and  100°  the  value  of  b  in  the  case  both 
of  copper  and  carbon  is  very  small.  A  determination 
of  the  coefficients  for  copper,  made  in  the  Physical 


95 


Laboratory  of  Cornell  University,  for  example,  yielded 
the  equation  :  t 

Rm"  =  Em'  (1  +  .00380  t  +  .00000047  t2).    (108) 

The  experiments,  which  covered  a  range  of  100°,  are 
in  close  agreement  with  the  results  published  by 
Matthiessen. 

In  the  above  equation,  for  t  —  100°,  we  have  bt2  = 
.0047.  If  we  neglect  the  coefficient  b  and  adopt  for  a 
the  mean  coefficient  between  o°  and  io~°,  we  may  write 
the  equation  in  the  simpler  form, 

fi™    =  ^m'  (1  +  .003847  t\  (109) 

which  will  give  results  agreeing  with  the  complete  form 
at  o°  and  ioc°  and  will  have  a  maximum  error,  at  50°, 
of  .0008. 

In  the  case  of  carbon  the  coefficient  b  may  also  be 
neglected  without  appreciable  error. 


R. 


FIG.    64. 

A  Carre  pencil,  measured  in  the  same  laboratory, 
gave  as  mean  coefficient  between  o°  and  100°,  the  value 
.000235. 

We  may,  therefore,  write  the  equation  for  this  variety 
of  carbon  as  follows  : 

R^  =  Ec'  (1  —  .000235  t).  (110) 

In  combining  copper  and  carbon  in  such  proportions 
that  the  resulting  resistance  shall  be  independent  of  the 
temperature,  the  equation  of  condition  must  be  satisfied. 
This  equation  will  be  satisfied  only  for  a  range  of 
temperatures  throughout,  which  b2t  is  negligible,  in  the 
case  of  both  substances. 

For  such  a  range  of  temperatures  we  have,  however, 
as  already  indicated, 

J3m»  =  Bm'  (1  +  am  t) 
R;  =£.'  (I -a,  t);  (111) 

when   ac   and   am  are  the   coefficients  -for   carbon   and 
copper  respectively. 

96 


Within  such  limits 


t) 


--  ac  t) 


m  c  m  am 

which  is  readily  reducible  with  a  sufficient  degree  of 
approximation  to  the  form  given  in  (103). 

A  convenient  form  for  such  a  resistance  is  shown  in 
Fig.  66.  It  consists  of  a  rod  of  carbon  about  twenty 
centimeters  in  length,  the  ends  of  which  have  been 
copper  plated  and  then  soldered  to  massive  copper  ter- 
minals. These  are  bent  at  right  angles  and  amalga- 
mated for  convenience  in  making  connections  by  means 
of  mercury  cups.  (See  Fig.  65.)  The  copper  compen- 


FIG.    65. 


sation  may  be  obtained  by  means  of  a  insulated  wire  of 
proper  resistance,  coiled  snugly  in  spiral  form  around 
the  rod  from  end  to  end.  A  glass  tube  protects  the 
apparatus  from  damage. 

Compensated  resistances  can  be  made,  in  a  variety  of 
other  forms,  according  to  the  materials  in  hand  and  the 
requirements  of  the  case.  A  very  simple  and  excellent 
form  consists  of  an  incandescent  lamp  with  german 
silver  wire  placed  in  series  or  in  multiple  with  the  fila- 
ment. This  form  is  of  much  higher  resistance  than 
that  depicted  in  Fig.  66,  and  will  carry  less  current.  It 
is  available,  indeed,  only  in  cases  where  the  heating 
effect  of  the  current  would  be  inappreciable. 


97 


The  use  of  a  compensated  resistance,  of  the  character 
first  described,  is  very  convenient,  since  it  does  away 
with  the  necessity  of  considering-  fluctuations  to  which 
an  uncompensated  conductor  will  be  subject  as  the 
result  of  the  heating  effect  of  the  current  flowing  in  it, 
and  of  temperature  disturbances  from  without.  In 
Lecture  VII,  it  has  been  shown,  however,  that  it  is 
entirely  practicable  to  determine  accurately  the  resis- 
tance of  a  metallic  conductor  when  carrying  current. 
By  the  application  of  the  methods  therein  described, 
especially  of  the  second  method  of  the  time  curve,  en- 
tirely satisfactory  calibrations  of  a  sensitive  galvanom- 
eter for  the  measurement  of  low  electromotive  forces 
may  be  obtained. 

An  important  example  of  the  use  of  sensitive  galvan- 
ometers in  the  determination  of  electromotive  force  is 
found  in  the  comparison  of  standard  cells.  The  gal- 
vanometer for  such  purposes  should  be  sensitive  to 
one  hundred-thousandth  of  a  volt,  and  its  figure  of 
merit  should  be  known  with  a  fair  degree  of  accuracy. 
The  conditions  of  such  determinations  involve  the  ready 
measurement  of  small  differences  of  potential  with  the 


FIG.   66. 

expenditure  of  as  little  current  as  possible.  The  usual 
method  of  proceedure  is  to  place  the  two  cells  which 
are  to  be  compared  in  opposition  to  each  other,  and  to 
use  the  galvanometer  ballistically,  a  sufficient  amount 
of  resistance  being  placed  in  circuit  to  place  its  deflec- 
tion to  a  suitable  quantity.*  One  of  these  two  cells  will 
usually  be  the  standard  with  which  others  are  to  be 
compared.  The  standard  cell  must  be  maintained  at  a 
constant  temperature,  viz.,  that  for  which  its  electro- 
motive force  has  been  previously  determined.  By  sub- 
stituting vsuccessively  the  various  cells  which  are  to  be 
compared,  these  being  placed  in  such  a  direction  as  to 
oppose  the  standard,  very  accurate  determinations  of 
the  ratios  of  their  electromotive  forces  to  that  of  the 
standard  may  be  obtained.  In  the  case  of  the  Clark 
cell  the  temperature  coefficient  has  been  made  a  matter 
of  careful  study  by  Clark,  Rayleigh,  Wright,  Helm- 
holtz,  Kittler,  and  more  recently  by  Carhart.  The  value 
of  this  coefficient,  according  to  the  different  observers, 

*  For  details  of  methods  of  testing,  see  Carhart ;  Primary  Batteries. 

98 


varies  through  a  considerable  range,  as  will  appear  from 
the  following  table  : 

Loss  per  degree 
Observer.  Centigrade. 

Clark1 0.0006 

Helmholtz2 0.0008 

Kittler3 0.0008  ' 

Rayleigh* 0.00077 

Wright5 0.00041 

Von  Ettingshausen6 0.00068 

Carhart7 0.000387 

The  construction  of  Rayleigh's  form  of  the  Clark  cell 
is  shown  in  Fig.  67.  Such  cells  give  constant  and  com- 
parable values  only  under  very  careful  treatment,  and 


ire  and  Zinc  Rod. 


U 9    — 


Glue. 

Cork 

Solntion  of  Zn  S  0, 

I/ff,  S  0,  (pasta.) 


20°  80°  40° 

FIG.    68. 


Pt  Wire. 


FIG.    67. 

even  when  handled  in  a  manner  which  ensures  their 
integrity,  they  possess  a  high  temperature  coefficient. 

The  source  of  the  variation  in  the  temperature  coffi- 
cient  of  Clark  cells,  has  been  clearly  pointed  out  by 
Carhart,  who  has  succeeded  by  modifying  the  cell  in 
such  a  way  as  to  reduce  the  coefficient  to  a  minimum 
value,  by  eliminating  one  of  its  elements,  the  variation 
in  the  concentration  of  the  solution  of  zinc  sulphate. 

This  done,  by  the  use  of  a  solution  which  is  saturated 
at  o°  (or  at  some  temperature  below  that  at  which  the 
cell  is  to  be  maintained)  the  variation  due  to  changes  in 

1.  Latimer  Clark;  Journal  of  the  Society  of  Telegraph  Engineers,  Vol.  7,  p.  53. 

2.  Von  Helmholtz;  Sitzungs  Berichte  der  Berliner  Akademi,  1882,  p.  26. 

3.  Kittler;  Wiedemann's  Annalen,  17,  p.  890. 

4.  Rayleigh  and  Sidgwick;  Proceedings,  Royal  Society,  17,  1884. 

5.  Wright;  Philos  Magazine,  5,  Vol.  16.  p.  25,  1883. 

6.  Von  Ettingshausen;  Zeitschrift  fur  Electrotechniker,  (Wien),  1884,  p.  i. 

7.  Carhart;  Primary  Batteries,  p.  93. 


99 


ERSITT 


the  density  of  this  electrolyte  vanishes  and  the  tempera- 
ture coefficient  falls  to  its  normal  value  (0.000387).  The 
coefficient  of  such  cells  can  be  expressed  graphically, 
as  function  of  the  temperature,  as  in  Fig.  68  or  by  the 
equation, 

Et=E^(l— 0.00038T(rf— 15)+0.0000005(£— 15)2        (113) 

The  Clark  cell  was  made  use  of  by  the  Chamber  of 
Delegates  of  the  Chicago  International  Congress  of 
Electricians  in  the  establishment  of  a  practical  unit 
of  electromotive  force.  Their  definition  was  as  follow  : 

"  The  International  volt  is  the  electromotive  force 
which  steadily  applied  to  a  conductor  whose  resistance 
is  one  international  ohm,  will  produce  a  current  of  one 
international  ampere,  and  which  is  represented  suffi- 
ciently well  for  practical  use  by  |£-|£  of  the  electromotive 
force  between  the  poles  of  electrodes  of  the  voltaic  cell, 
known  as  Clark's  cell,  at  a  temperature  of  15°." 

If  one  is  in  posession  of  two  Clark  cells,  the  tempera- 
ture coefficients  of  which  are  well  known,  these  may  be 
used  in  combination  as  a  means  of  procuring  a  standard 
of  very  small  electromotive  force.  By  maintaining  the 
two  at  slightly  different  temperatures  ;  and  placing 
them  in  circuit,  back  to  back,  they  may  be  made  to  pro- 
duce a  perfectly  definite  and  very  small  electromotive 
force  by  their  differential  action,  and  in  this  combination 
may  be  used  for  the  purpose  of  calibration.  A  much 
more  convenient  source  of  small  electromotive  force 
which  serves  as  an  admirable  secondary  standard,  is  a 
thermo-element  of  copper-iron,  copper-german  silver  or 
platinum-iridium,  according  to  the  range  desired. 

It  is  not  possible,  in  constructing  such  a  thermo-ele- 
ment out  of  the  materials  ordinarily  attainable,  to  secure 
a  standard  of  electromotive  force  which  is  absolute  in 
the  sense  of  giving  the  same  values  in  a  case  of  different 
individual  elements,  since  the  differences  between  the 
various  members  of  a  series  of  such  elements,  even 
when  these  are  constructed  as  nearly  as  possible  from 
like  materials,  will  be  found  to  be  considerable.  Once 
constructed,  however,  such  thermo-elements  are  not 
subject  to  marked  fluctuations  in  their  character.  It  is 
possible,  therefore,  to  make  a  thermo-element  and  to 
calibrate  it  once  for  all.  It  may  then  be  used  as  a  sec- 
ondary standard  of  small  electromotive  forces  ;  the 
only  further  precautions  being  those  involved  in  bring- 
ing the  two  junctions  to  a  known  temperature  difference 
and  maintaining  them  there.  The  temperatures  most 
easily  maintained  are,  of  course,  those  of  melted  ice  and 
of  steam  at  normal  pressure.  The  arrangement  of  such 
a  standard  is  described  in  a  previous  lecture. 

100 


LECTURE   IX. 

THE  USE  OF  THE   GALVANOMETER    FOR   THE    MEASURE- 
MENT OF  TEMPERATURE. 

The  types  of  instrument  iiseful  for  the  determination 
of  temperatures  are  those  of  maximum  sensitiveness, 
described  in  Lecture  VI.  In  thermometric  work  the 
galvanometer  is  used  : 

(1)  In  measuring  changes  in  the  resistance  of  a  wire 
by  the  method  of  fall  of  potential. 

(2)  In  the  Wheatstone  bridge. 

(3)  In  circuit  with  a   thermo-pile  or  thermo-electric 
couple. 

The  first  method  has  a  very  wide  range  of  useful- 
ness. If,  for  example,  temperatures  are  to  be  deter- 
mined in  a  locality  in  which  the  mercury  thermometer 
cannot  be  used,  a  coil  of  pure  copper  wire  may  be  pre- 
pared, the  resistance  of  which  at  a  known  temperature 
is  accurately  determined  once  for  all,  as  also  its  tem- 
perature coefficient  for  the  entire  range  of  temperatures 
under  consideration.  This  coil  having  been  placed  in 
the  locality  for  which  the  temperatures  are  to  be 
measured  is  connected  with  the  galvanometer  by  line 
wires  of  negligible  resistance.  A  comparison  coil,  the 
resistance  of  which  should  be  as  nearly  as  is  convenient 
the  same  as  that  of  the  temperature  coil,  is  placed  in  a 
bath  of  constant  temperature.  This  comparison  coil 
should  be  constructed  of  material  having  as  small  a 
coefficient  temperature  as  possible,  or  it  may  be  com- 
pensated by  the  methods  described  in  Lecture  VIII. 
The  temperature  coil,  Rt,  and  the  comparison  coil,  RC, 
are  placed  in  series  with  some  suitable  source  of  current 
(B),  and  the  circuit  is  permanently  closed.  The  gal- 
vanometer is  placed  alternately  in  shunt  with  the  two 
coils  (see  Fig.  69)  and  the  ratio  of  the  deflections  thus 
obtained  gives  the  resistance  of  the  temperature  coil  in 
terms  of  that  of  the  other.  As  has  already  been  pointed 
out  in  the  lecture  just  referred  to,  this  method  of  alter- 

101 


nate  deflections  eliminates  the  fluctuations  in  the  figure 
of  merit  in  the  galvanometer  and  affords  a  means  of 
ready  and  accurate  measurement  of  the  variations  of 
resistance  of  the  temperature  coil  and  so  indirectly  of 
the  changes  of  temperature  which  occur  in  the  locality 
in  which  it  is  placed.  When  it  is  desired  to  integrate 
the  temperatures  existing  throughout  a  region  of  con- 
siderable extent,  the  wire  instead  of  being  wound  into 
a  coil  is  carried  through  the  entire  region  to  be  studied. 
If,  for  example,  we  desire  to  know  the  average  temper- 
ature of  a  standard  bar,  the  length  of  which  is  to  be 
measured,  the  wire  may  be  wound  around  the  bar 
longitudinally  a  sufficient  number  of  turns  being  made 
to  give  the  desired  resistance  to  the  temperature  coil. 


FIG.  69. 

This  method  has  been  found  to  be  very  satisfactory  in 
determining  the  coeffiicient  of  the  expansion  of  such 
bars.* 

Another  example  of  the  application  of  this  method  is 
found  in  the  determination  of  the  average  temperature 
of  the  phosphor-bronze  suspension  wire  of  the  swing- 
ing coil  described  in  Lecture  IV.  This  wire  was 
stretched  vertically  through  a  distance  of  two  meters, 
and  it  formed  the  suspension  of  the  coil.  All  attempts 
to  get  its  temperature  by  means  of  mercury  thermo 
meters  proved  very  unsatisfactory,  but  by  means  of  a  No. 
40  copper  wire  carried  several  times  the  entire  length 
of  the  suspension  tube,  very  good  results  were  ob- 
tained.f 

*  See  the  report  of  Joseph  Le  Conte  upon  the  coefficient  of  expansion  of  a  standard 
meter  made  by  the  Societe  Genevoise  ;  reports  of  the  Physical  Laboratory  of  Cornell 
University,  1891. 

+  See  N.  H.  Genung's  Thesis  on  the  Electro-Chemical  Equivalent  of  Silver,  Cor- 
nell University  Library. 

102 


Still  another  illustration  of  the  application  of  this 
method  of  measuring  temperatures  is  afforded  by  the 
experiments  of  Messrs.  Child,  Quick  and  Lanphear* 
upon  the  distribution  of  temperature  along  a  copper 
bar,  one  end  of  which  was  heated.  The  object  of  the 
experiment  was  to  determine  the  thermo-conductivity 
of  the  bar.  For  this  purpose  the  distribution  of  temper- 
atures after  the  bar  had  reached  its  final  condition  was 
necessary.  A  collar  consisting  of  a  single  layer  of  very 
fine  insulated  copper  wire  fitting  closely  around  the 
bar  made  it  possible  to  measure  the  temperature  at 
different  points  throughout  the  entire  length  of  the 
latter  with  great  accuracy.  In  this  case,  however,  the 
measurements  were  made  by  the  method  of  the  Wheat- 
stone  bridge. 

The  best  form  of  apparatus  for  the  application  of  this 
method  is  the  slide  bridge.  In  Fig.  70,  which  shows  the 
connections,  A  B  is  the  slide  wire  of  platinum  iridium,  p. 


FIG.  70. 


the  sliding  contact,  Rt  and  RC  are  the  temperature  coil 
and  compensated  resistance  respectively,  while  rv  and 
rz,  taken  together  with  the  two  parts  of  the  slide  wire, 
are  the  other  arms  of  the  bridge. 

With  this  arrangement  of  the  apparatus  the  temper- 
ature may  be  conveniently  expressed  in  terms  of  the 
position  of  the  sliding  pointer  p. 

The  resistance  of  copper  is  well  adapted  for  the  meas- 
urement of  temperature,  since  its  changes,  through  a 
very  wide  range,  are  nearly  proportional  to  the  temper- 
ature. It  is,  indeed,  only  above  200°  and  below  io<.° 
that  the  change  in  the  coefficient  is  such  as  to  introduce 
grave  errors. 

The  coefficients  of  different  wires  vary  somewhat, 
however,  in  absolute  value,  so  that  the  specimen  of 
which  the  temperature  coil  is  to  be  made  should  always 

*  Physical  Review,  Vol.  II. 

103 


be  calibrated.  Many  samples  of  modern  commercial 
copper  show  a  coefficient  of  resistance  greatly  in  excess 
of  Matthiessen's  value  for  the  pure  metal.  Kennelly 
and  Fessenden*,  for  example,  in  1893  found  for  a  copper 
wire  0.004065,  with  variations  from  that  value  at  27.8° 
of  0.000058  and  at  255°  of  -f-  0.000005. 

Values  lying  between  .0041  and  .0042  are  frequently 
observed  in  modern  practice.     Thus  Dewar  and  Flem- 


ing found  for  low  temperatures  0.00410  ;  Quick  and 
Lanphear,  between — 38.6°  C.  and  -j-  i.5°C.  obtained 
0.004147  ;  Cailletet  and  Bouty's  value  was  higher  than 
any  of  these,  viz.:  —  0.00423. 

The  very  nearly  constant  value  of  the  resistance  co- 
efficient  is     shown    graphically    in    Figs.    71    and   72, 


^Physical  Review,  vol.  x. 


104 


which  are  plotted  from  the   results   of   Kennelly  and 
Fessenden  and  of  Quick  and  Lanphear  respectively. 

These  curves  are  plotted  to  different  scales  and  they 
apply  to  different  specimens  of  copper,  not  of  the  same 
quality  as  to  the  coefficient  of  resistance.  The  meth- 
ods of  calibration  also  were  distinct.  Both  are  straight 
lines,  however,  indicating  an  unvarying  coefficient 
through  wide  range  of  temperatures,  including 
the  important  interval  below  40°  C.,  which 

cannot     be     reached     with      mercury     thermometers. 


The  calibration  of  a  temperature  coil,  should  always 
be  made  under  conditions  as  nearly  as  possible,  identical 
with  those  under  which  it  is  to  be  used.  Otherwise  the 
temperature  lag,  of  the  coil  with  reference  to  the  body 
the  temperature  of  which  is  to  be  determined,  (or  vice 
versa)  will  introduce  errors  which  will  always  be  ap- 
preciable excepting  when  temperatures  have  become 
strictly  stationary,  and  which  may  sometimes  rise  to  un- 
suspected size. 


105 


The  nature  of  this  error  in  a  typical  case,  that  in 
which  the  temperature  of  a  copper  bar,  was  to  be  meas- 
ured by  means  of  a  collar  of  fine  wire  surrounding  it,  is 
indicated  in  Fig.  73  (from  determinations  by  the  observ- 
ers just  cited).  In  this  diagram  abscissas  are  tempera- 
tures of  the  bar,  as  indicated  by  a  thermometer,  the 
bulb  of  which  was  immersed  in  a  mercury  capsule  with- 
in the  body  of  the  metal,  while  ordinates  are  readings 
on  the  slide  bridge.  The  bridge  gives  relative  readings 


of  the  temperature  of  the  collar.  The  crosses  and  cir- 
cles show  the  temperatures  of  the  bar  at  which  the 
bridge  readings  were  the  same,  for  rising  and  falling 
temperatures  respectively.  The  curves  of  heating  and 
cooling  to  which  these  apply  were  very  nearly  identical 
and  the  temperature  difference  at  any  point,  between 
an  observation  and  the  median  line,  gives  the  lag  (posi- 
tive or  negative).  It  will  be  noted  that  in  this  case  the 
error  of  assuming  the  temperature  of  the  collar  to  be 
that  of  the  bar  would  have  been  about  one  degree. 

106 


For  very  high  temperatures,  copper  is  not  available 
for  electrical  thermometry  and  consequently  many 
attempts  have  been  made  to  substitute  platinum  for  that 
metal.  The  law  of  resistance  for  platinum,  must  how- 
ever be  determined  for  each  specimen  and  this  calibra- 
tion is  a  matter  of  extreme  difficulty. 

A  comparison  of  the  various  formulae  proposed  by 
Matthiessen,  Siemens,  and  Benoit,  for  determining  tem- 
peratures from  the  resistance  of  platinum  was  made 
some  years  ago  by  the  writer.*  His  curves  showing  the 


2400* 

"                      ^a"  i   '  *"*      ^*/  /&*    ' 
^^  '  /  <k*         '  7     /'2* 

2200' 
2000' 
1800° 

:    l^^?" 

1600° 

\kj/  / 

»  i    i>  /     * 

1400° 

!!///'/ 

1200* 
1000C 
SOO3 

I  III// 
'•  i  .'•'.'/ 

'   IW 

600' 

/// 
/"' 

400° 

200e 

] 

\        2345578 

FIG.  74. 

divergent  character  of  the  results  which  would  be  ob- 
tained by  the  application  of  their  formulae,  are  given 
in  Fig.  74. 

Another  method,  applicable  alike  to  very  high  and  to 
very  low  temperatures,  is  that  of  the  thermo-element  of 
platinum — platinum-iridium.  The  same  caution  must 
be  observed,  however,  with  reference  to  commercial 
specimens  as  in  the  method  previously  described. 

*  Am.  Journal  of  Science,  vol.  22,  p  363. 

107 


Platinum  wire  purchased  from  leading  dealers  in  this 
country,  and  supposed  to  be  pure,  gave  in  the  hands  of 
the  writer  curves  of  E.  M.  F.  and  temperature  of  the 
character  shown  in  Fig.  75,  one  being  concave  and  the 
other  convex  to  the  base  line.  These  wires  were  com- 
bined with  the  same  quality  of  platinum-iridium  in  the 
construction  of  the  thermo  elements. 

It  will  be  seen  that  a  deflection  which  corresponds  to 
600°  C.  in  the  one  case  would  be  reached,  with  the  other 
thermo  element  at  900°  C. 

Barus,  in  his  exhaustive  research  upon  the  measure- 
ment of  high  temperatures,  found  similar  peculiarities 


400°  600° 

FIG.    75. 


in  commercial  wires,  but  when  he  used  an  element 
composed  of  standard  materials  he  obtained  a  curve 
which  is  very  nearly  straight  through  a  wide  range  of 
temperatures. 

Excellent  results  have  also  been  reported  with  this 
couple  at  very  low  temperatures,  but  it  should  be  noted 
that  some  commercial  samples  give  reversal  at  a  tem- 
perature slightly  above  o°. 

The  methods  thus  far  described  can  be  pursued  with 
galvanometers  of  medium  delicacy  ;  when,  however, 
we  come  to  the  exceedingly  small  temperature  differ- 

108 


ences  with  which  the  student  of  radiant  heat  has  to 
deal,  instruments  of  the  highest  sensitiveness  are  es- 
sential. 

Nearly  all  measurements  in  this  domain  are  of  a  rel- 
ative character  ;  Knut  Angstrom,  however,  (1893)  has 
described  a  bolometric  method  by  means  of  which  ab- 
solute determinations  may  be  made,  and  the  results 
expressed  in  gram-calories  per  second  per  c  m9. 

The  principle  of  Angstrom's  method,  as  described  in 
his  paper,*  is  briefly  as  follows  :  "  Given  two  thin  strips 
of  metal,  A  and  B  (Fig.  76),  which  are  as  nearly  as  pos- 
sible identical.  The  sides  of  these,  which  are  exposed 
to  the  source  of  heat,  are  blackened,  and  the  strips  are 
arranged  in  such  a  way  that  it  is  possible  to  determine 
accurately  when  they  are  of  the  same  temperature. 
These  strips  are  so  placed  that  a  current  of  any  desired 
strength  can  be  sent  through  them.  If  one  of  them, 
for  example,  A,  is  exposed  to  the  source  of  heat,  while 


FIG.    76. 

B  is  protected  by  a  screen,  wre  may  restore  the  balance 
of  temperature  which  has  been  disturbed  by  the  ab- 
sorption of  heat  on  the  part  of  A,  by  sending  a  current 
of  proper  intensity  through  B.  When  the  temperatures 
are  the  same,  then  the  amounts  of  energy  which  A  and 
B  have  received  are  equal  to  one  another.  Let  /  and  b 
be  the  length  and  width  of  the  strips,  r  the  resistance 
of  the  same,  and  /  the  current.  Then  since  the  heat 
absorbed  by  A  is  the  equivalent  of  that  produced  by 
the  electric  current  in  b,  we  may  write 


where  q  is  the  radiant  energy  received  by  a  unit  of  sur- 
face. In  order  to  counteract  the  inequality  of  the 
strips,  they  are  interchangeable,  B  being  illuminated 
and  the  current  sent  through  A. 

*Angstrora  ;    Trans.  Royal  Soc.  of  Sc.  Upsala,  1893  ;  also  Physic.  I  Review,  vol.  i, 
P-  365- 

I09 


"  In  the  practical  application  of  this  principle  of  com- 
pensation, one  may  follow  various  methods.  The  equal- 
ity of  temperature  may  be  determined  in  a  variety  of 
ways.  If  thermo-elements  are  used  for  this  purpose, 
one  needs  a  sensitive  galvanoscope,  and  the  measure- 
ment of  the  current  can  be  made  upon  an  instrument  of 
ordinary  delicacy.  It  is  possible,  however,  to  carry  on 
the  investigation  without  any  difficulty,  using  only  one 
galvanometer,  a  plan  which  I  pursued  in  the  case  of  the 
first  apparatus  which  I  constructed.  Fig.  77  gives  a 
diagram  of  the  connections. 

"The  metallic  strips,  A  and  B,  cut  simultaneously  from 
two  thin  sheets  of  platinum  laid  one  upon  another,  are 
0.154  cm.  wide  and  1.80  cm.  in  length.  They  are  black- 
ened in  the  usual  way  upon  the  side  exposed  to  radia- 
tion, and  are  mounted  side  by  side  in  a  frame  of  ebonite. 


FIG.    77- 

This  frame  is  inserted  in  a  tube.  Upon  the  back  of  the 
strips  are  laid  two  exceedingly  thin  leaves  of  mica,  up- 
on which  very  minute  junctions  of  copper  and  German 
silver,  are  inserted.  To  hold  the  mica  and  the  therm o- 
j  unctions  together,  I  used  marine  glue  in  as  thin  a  layer 
as  possible. 

"  One  of  the  strips  is  subjected  to  radiation  ;  the  cir- 
cuit is  then  closed  through  the  other  one  ;  the  thermo- 
element is  brought  into  circuit  with  the  galvanometer 
by  means  of  the  commutator  ;  the  sliding  contact  is  ad- 
justed until  the  galvanometer  stands  at  zero  ;  the  com- 
mutator is  then  reversed,  and  the  strength  of  the  cur- 
rent used  in  heating  the  strip  is  determined.  The 
switch  is  then  reversed,  the  shutter  is  placed  in  front  of 
the  other  strip,  and  the  setting  and  current  measure- 


ment  are  repeated.  There  are  two  other  ways  in  which 
one  can  use  this  apparatus  for  the  measurement  of  rad- 
iation, viz  :  — 

First  Variation  of  the  Method,  —  One  of  the  strips,  for 
example,  A,  is  exposed  to  radiation,  while  the  other  is 
screened.  One  notes  the  deflection  of  the  galvanom- 
eter, which  becomes  constant  in  about  fifteen  seconds. 
A  is  then  also  screened,  and  by  means  of  the  current  is 
brought  to  the  same  temperature  to  which  radiation 
had  previously  brought  it.  The  strength  of  the  current 
producing  this  rise  of  temperature  is  then  measured. 
The  advantage  of  this  arrangement  is  that  the  same 
strip  is  warmed  by  means  of  radiation  and  then  through 
the  agency  of  the  current,  so  that  it  is  not  necessary  to 
be  so  painstaking  as  regards  the  identity  of  the  two 
strips. 

Second  Variation  of  the  Method.  —  One  of  the  strips,  for 
example,  A,  is  exposed  to  radiation,  the  other,  B,  is 
screened  ;  the  deflection  of  the  galvanometer  is  observed 
after  the  thermo-current  has  become  constant.  The 
quantity  of  heat  which  A  has  received  is  calculated  from 
Newton's  law  of  cooling. 


where  6  is  the  difference  of  temperature  indicated  by 
the  galvanometer,  and  k  is  the  constant  of  cooling  of 
the  strip.  If  we  now  send  a  current  through  A,  without 
making  any  change  of  conditions,  the  difference  of  tem- 
perature will  become  greater  still.  This  will  be  indic- 
ated by  the  resulting  deflection  0^  The  strength  of  the 
current  producing  this  additional  heating  effect  may  be 
measured  in  the  manner  already  described." 

Galvanometer,  suitable  for  investigations  are  easily 
constructed,  following  the  general  principles  of  design 
given  in  Lecture  VI. 

In  undertaking  any  extended  bolometric  work,  or 
other  research  of  great  delicacy,  the  galvanometer 
should  be  especially  constructed  for  the  particular  pur- 
pose which  it  must  serve.  The  question  of  the  period 
of  oscillation  is  sometimes  an  important  one  ;  on  account 
of  local  magnetic  disturbances.  The  proper  adapt- 
ation of  the  resistance  to  the  circuit  in  which  "the  instru- 
ment is  to  be  used  is  of  especial  significance. 

Very  often,  the  experimenter  with  the  bolom- 
eter is  compelled  to  use  his  galvanometer  under  condi- 
tions of  the  highest  attainable  delicacy.  Then,  if  he  is 
to  obtain  readings  which  possess  value,  his  patience  and 
skill  will  be  taxed  to  the  utmost.  For  the  difficulties 
which  arise  in  such  researches  no  general  prescription 
can  be  written.  The  vagaries  of  the  needle  under  the 

in 


combined  influences  of  diurnal  magnetic  drift,  local  mag- 
netic disturbance,  thermo-electric  differences,  and  the 
obscure  thermal  fluctuations  to  which  the  bolometer 
and  its  accessories  are  subject  must  be  studied  as  they 
arise,  and  overcome.  These  causes  can  be  detected, 
and,  after  due  experience,  ingenuity  and  tireless  pat- 
ience will  bring  their  effects  under  control  ;  but  more 
remote  sources  of  disturbance  are  perpetually  at  work 
against  the  bolometrist  :  A  moderate  gale  of  wind,  an 
auroral  display,  almost  too  faint  to  be  visible,  even  a 
storm  in  the  solar  atmosphere,  will  drive  him  from  his 
seat  at  the  reading  telescope  in  despair. 

These  are  difficulties  to  be  surmounted  only  by  him 
who  can  wait  ;  he  it  is,  alone,  who  may  enter  the  realms 
of  research  which  lie  along  the  very  boundary  line  of 
human  attainment  ;  in  his  hands  alone  do  we  see  the 
highest  performance  of  that  remarkable  instrument,  the 
galvanometer. 


FWIS. 


ME 

ERSITT 


112 


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